A well-known feature of the Samoyedic languages is a split development of Proto-Uralic *u. The standard analysis (as first proposed, IIUC, by Janhunen 1981) is that this occurred depending on the original stem type.
- *u becomes *ə before original 2nd syllable *-a. E.g.
- *kupsa- > *kəptå- ‘to extinguish’
- *lupsa > *jəptå ‘dew’
- *muka > *məkå ‘back’
- *muna > *mənå ‘egg’
- *mud₂a- > *məjå ‘earth’
- *tud₂ka > *təjkå ‘tip’ 
- *u generally remains before original 2nd syllable *-ə. E.g.
- *kunśə > *kunsə ‘urine’
- *suksə > *tutə ‘ski’
- *suxə- > *tu- ‘to row’
- *tulə > *tuj ‘fire’
- *ujə- > *uə- ‘to swim’
I have not seen the details of how this happened discussed often, but it all seems straightforward enough. There are at least three reasons to believe that the former of these has been a conditional development, and the latter unconditional.
- The most obvious reason is identity: “*u > *u” is not even a change, but the absense of one.
- The second is the nature of the conditioning. Words of the 1st set are all united by the presence of *-a. Words of the 2nd do not fare as well, since 2nd syllable *-ə has been regularly lost after original light syllables / single consonants; as in ‘fire’. (I actually suspect that this change needs to be dated as very early, already prior to most individual developments of Samoyedic.) This does have the immediate result of rendering the 1st syllable closed — but this is further disrupted by the loss of medial velar consonants as in ‘ski’, ‘to row’. Moreover, *u > *ə occurs in closed syllables followed by *-a all the same; as in ‘to extinguish’, ‘dew’, ‘tip’.
- Finally, phonetically, the development *u > *ə would be well explainable as a simple assimilation development: when followed by the open, [-ATR] vowel *a, *u also becomes [-ATR] (that is: *ʊ), and later drops its labiality to yield *ə.
Observe though that nothing in the previous scenario is dependent on *u being a back vowel. So we should perhaps expect to see something similar also going on with the PU close front vowel *i. And indeed, examples of *i > *ə are known. So far, these have however been explained by rather different means.
- *śilmä > *səjmä ‘eye’ is explained by Janhunen (1981) and Sammallahti (1988) as dissimilation caused by the preceding *j.
- *itä- > *ətä- ‘to appear’, *ipsɜ(-) > *əptə(-) ‘(to) smell’ have been explained as being conditioned by the lack of a word-initial consonant by Aikio (2002).
- Helimski (1993) lists a number of further cases, most interestingly including *pišä > *pətä ‘gall’. He notes that the conditions of the change seem unclear. 
(cf. my Bibliography page for the reference details)
Examining the words above as a group, the possibility of a more economic analysis seems self-evident to me: to unite *u > *ə and *i > *ə as a single “a-umlaut” affecting close vowels. Reduction and lowering of close vowels is, after all, almost always a process that operates symmetrically, without regard for vowel frontness/backness.  I also find it phonetically highly unclear how the absense of a word-initial consonant could trigger vowel reduction! Aikio mentions also a third example which clearly must be considered an original *ə-stem: *əm- ‘to suck’ < PU *imə- (cf. Finnish ime- ‘id.’). I wonder if this might be, rather than a regular development, due to the influence of the homonymous PSmy root *əm- ‘to eat’.
Also ‘to smell’ has a stem vocalism issue: Janhunen & Sammallahti reconstruct this as *ipsi (= *ipsə according to my transcription). But here the cognates do not seem to unambiguously point to an original non-close stem vowel. Several examples are known where a PU open stem vowel has been reduced in Samoyedic; and it even seems to be the case that this is especially frequent after consonant clusters.  In Samic, the only other diagnostic branch where reflexes of this root survive, both a noun *(h)ëpsë ‘smell’ < *ipsə, and a verb *(h)ëpsē- ‘to smell’ < *ipsä- are found. Also Mordvinic would normally offer evidence, but the proposed Moksha cognate /opəś/ ‘smell’ is so divergent that I wonder if it has any relation to this PU root at all. Thus the choice of which root shape to consider more original cannot be resolved, and *ipsä remains a viable option. It would even be possible to reconstruct an alternation *ipsə ‘smell’ ~ *ipsä ‘to smell’ already to Proto-Uralic. (In this case I’d have to assume that in early pre-Samoyedic, this lexeme showed an alternation *ipsə ~ *ɪpsä-, which was then levelled to uniformly *ɪpsə(-).)
It is also possible to include here the 1st and 2nd person singular pronouns: PSmy *mən, *tən. My proposed soundlaw *i-ä > *ə would mean these being fairly regularly compareable to the proto-forms *minä, *tinä indicated by most of the Uralic family: Finnic, Mari (2PS only), Permic, Hungarian (2PS only) and Khanty (1PS only). By contrast Samic and Mordvinic suggest back-vocalic *mun, *tun, but these show no evidence of a former 2nd syllable that could have triggered umlaut. Admittedly, neither do the Samoyedic words, but in them a reduction of PU *-ä to PSmy *-ə could again have applied. There is also at least one precedent for the loss of such a secondarily reduced stem vowel: *puna- >> *pən- ‘to plait’. (Medial *-n- in both this and the pronouns seems likely to be an accidental similarity.) Assuming irregular reduction in function words does not seem like a major problem.
There is another piece of evidence in support of this derivation as well. As has been shown by Helimski (1993), whether any given instance of PSmy *ə derives from earlier *i or *u is still indicated by the declension harmony class, especially in Nganasan, and recently Janhunen (2013)  has noted that the personal pronouns indeed align as originally front.
As an aside: if also the Samoyedic pronouns derive from *minä, *tinä, we have the additional nice consequence that the forms *mun, *tun can be considered a common Samic-Mordvinic innovation. This would eliminate the issue of two distinct pronoun sets being reconstructible to Proto-Uralic, yet no trace of a distinction between these having been attested in any Uralic language.
There is also one piece of good counterevidence for the previously assumed conditioning for *i > *ə: *i ‘top’, which seems to derive via earlier *ij from an East Uralic *ilə- ‘up, over’ (cf. Mansi *äl-, Khanty *eeL-; contrast with *wülä- indicated by West Uralic *ülä-, Mari *wü̆l, Permic *vɨl-). Despite both a following *j and a lack of a word-initial consonant, and even a status as a short function word, it has not reduced to **ə-.
So, summing up, I would suggest that the usual development of *i has been exactly parallel to that of *u:
- *i > *ə before original 2nd syllable *ä.
- *ipsä- > *əptə- ‘to smell’
- *itä- > *ətə- ‘to appear’
- *minä > *mən ‘I’
- *pišä > *pətä ‘gall’
- *śilmä > *səjmä ‘eye’
- *tinä > *tən ‘thou’
- *i > *i before original 2nd syllable *ə. E.g.
- *ilə > *i ‘top’
- *nimə > *nim ‘name’
- *ńimə- > *ńim- ‘to suck’
- *pid₁ə > *pir- ‘tall’
Given that the overall amount of data we have of the development of PU *i in Samoyedic is fairly limited, I feel that being able to root its development in processes that can be confirmed with other data (i.e. the development of *u) is a good step up from explaining its development with a grab-bag of phonetically ill-defined processes based on two or three examples.
There still remains some evidence for close vowel reduction also in PU *ə-stem roots.
Although I have just noted how phonetically unmotivated sound changes, if based on limited data, might well turn out to be illusions based on accidental patterns in the data, I will venture one hypothesis for an additional condition for reduction: before PU *-ŋ-. There are two candidates for this:
- *piŋə ‘tooth’ > *pəj ‘stone; flint’?
Although most of this root’s reflexes have the primary meaning ‘tooth’, the rather specific meaning ‘flint’ can be also found in Finnic, and it may have also been present already in Proto-Uralic. In Samoyedic the word seems to have undergone semantic drift, first loosing its anatomic meaning, and then becoming the neutral word for ‘stone’ across much of the subfamily. — The development *ŋ > *j is unclear, although the same seems to be found in *kuŋə (or *këŋə?) > *kïj ‘moon’.
- *suŋə > *təŋə ‘summer’?
Here the retention of *-ə after an open syllable is very odd, but per the West Uralic evidence it is quite clear that an original *ə-stem must be reconstructed; cf. e.g. Finnish suvi ‘summer’, Northern Sami sagŋat ‘to thaw’.
This still does not exhaust the cases of inherited reduced vowels in Samoyedic, though. At least the following can also be noted:
- PU *kulkə- > PSmy *kəj- ‘to go’?
This may seem to recapitulate the alleged soundlaw *i > *ə / _j; but there are a number of good counterexamples against assuming this to have been regular also for *u, e.g. *ulkə > *uj ‘pole’, *tulə > *tuj ‘fire’. I suspect that the explanation could rather involve how *kulkə- seems to be an early loan from PIE *kʷelH- (or a pre- or para-IE *kulH-) ‘to turn, to go’. Perhaps the Samoyedic word has been acquired from a slightly different source than its apparent cognates. One form that would fit well here is Tocharian *kɨl-.
- PU *küńəl(ə) > PSmy *kəńələ ‘tear’?
Presumably a case of interference from PU *kuńa- > PSmy *kəńə- ‘to close eyes’.
- PU #pid₂mətä > PSmy *pəjmətä ‘dark(ness)’?
The PU reconstruction is tentative; the only other cognates are Finnic *pimedä, Udmurt /peĺmɨt/, Komi /pemɨd/. It might be possible to fix things if we were to start from something like *pid₂mä-, and to assume that the adjectival suffix on showcase here (perhaps the only case where it can be explicitly reconstructed already for PU) has originally been plain agglutivative *-tA, and not root-vowel-replacing *-ətA. In Finnic, quite a few alternations between CVCA noun/verb roots and CVCe-dA adjectivizations are found, but perhaps we should not project this too far back, and instead assume secondary vowel reduction in Finnic at some time-depth. The Permic reflexes would appear more regular in this case as well, as the other examples of the development *i > /e/ seem to be restricted to *ä-stem roots.
There are also some Proto-Samoyedic roots with apparent *i whose Uralic etymologies appear to imply original *-ä. I suspect these however have a different origin. In the Samic cognates of these, *i rather than the usual *ë appears. Similarly, the Khanty cognates have *ii rather than the usual *ee.
- Samic *imē ~ Khanty *iimə ~ PSmy *imɜ ‘grandmother’
- Samic *cicē ~ Khanty *čiinč ~ Selkup /čičik/ ‘small bird’
- Samic *ńińčē ~ Selkup /ńipsə/ ‘teat’
Contrast *śilmä > Samic *čëlmē, Khanty *seem ‘eye'; *ipsä- > Samic *ëpsē-, Khanty *eepəɬ- ‘to smell’.
It is unclear where this correspondence could originate: Samic *i-ē and Khanty *ii do not have any known regular origin. There are several words where Khanty *ii appears to derive from older *e though, particularly next to semivowels (Kh *liiɬ ‘breath, soul’ ~ Finnic *lewlü ‘steam’ < PU *lewlə?; Kh *niiŋ ‘girl’ ~ Finnic *nejtej ‘id.’ < PU *nejə-?) which could suggest proto-forms such as *čej(ɜ)nčä, *ejmä here.
It is also the case that a merger of *e and *i has occurred in most of Samoyedic, with the exception of Nganasan, where *e > /ɨ/ (e.g. *wetə > /bɨʔ/ ‘water’). But a second-degree exception to this applies: *e still ends up as Ng. /i/ when before /i/ or /ü/, and this could apply to ‘grandmother': Ng. /imi/. In Proto-Samic, in turn, *e before *-ä would yield *ea, but the structure *-eaj- appears to be mostly absent (with the exception of *peajvē ‘day’, which is however an irregular reflex of older *päjwä).
Regardless of if this is on the right track for explaining these words, it should be clear that given these sets of “regularly aberrant” reflexes, positing PU *čičä, *imä cannot be justified (and these comparisons are indeed absent from the stricter PU wordlists of Janhunen and Sammallahti). If so, there seem to be no cases where PU *i before 2nd syllable *ä would be retained as Samoyedic *i.
There is a proposal that might be worth a mention here, though. Tibor Mikola, in the monograph Studien zur Geschichte der samojedischen Sprachen (2004, Studia Uralo-Altaica 45), considers reconstructing distinct *ɪ, *ʊ already for Proto-Uralic, and their merger with *i, *u on the Finno-Ugric side. He seems to have picked this up from the idiosyncratic PU reconstruction of Katz, which I’ve mentioned previously. In principle words like ‘grandmother’ above could then allow contrasting the vocalisms *ɪ-ä and *i-ä. I am skeptical however, since the assumed FU development *ɪ >*i is typologically backwards, and this line of explanation appears to depend on the belief of a special status of Samoyedic within Uralic.
Notes on the non-cardinal close vowels
Anyway, if *u-a and *i-ä > *ə are regular developments in Samoyedic… what about the other close vowels, then?
I still hold that the non-close back unrounded, “eighth” vowel of Proto-Uralic should be reconstructed as mid *ë, not as close *ï. It should then be possible to simply date the emergence of Proto-Samoyedic *ï, in words of the type *mëksa > *mïtə ‘liver’, as later than the reduction of *u and *i. This might carry some consequences, but more on those at another time.
The case of *ü is less clear. This was unambiguously a close vowel in PU. Yet, the scarce Samoyedic examples we have of this vowel in *ä-stems display *i, not *ə:
- *d₂ümä > Smy *jimä ‘glue’
- *küčä > Smy *kitä ‘birch bark vessel’ (with irregular *-t-)
- perhaps: *lüwä- > Smy *jiwä- > Selkup /čü-/ ‘to shoot’ 
(NB the loss of rounding is not an issue: also in *ə-stems, *ü usually merged with *i, e.g. *tütkə- > *titə- ‘to open’, *nüd₁ə > *nir ‘handle’.)
Some sort of palatal cheshirization could be considered. Suppose that *ɪ, as resulting after the first stage of a-umlaut, managed to keep its timbre here, thanks to the word-initial *j and *k (the latter may have had a palatal allophone *[c] already in Proto-Samoyedic), and proceeded to climb back to *i? Still, a following *j in *səjmä ‘eye’ does not appear to have the same effect, so I am hesitant.
Alternately, perhaps this can be related to other ways in which *ü appears to not follow the example of *i and *u. Two issues come to mind:
- In Khanty, as mentioned, the default reflex of *i is tense *ee. *u is reflected as lax *o before original consonant clusters (*suksə > *ɬok ‘ski’), but tense *oo before original single consonants (*muna > *moon ‘egg’). Yet *ü is only ever reflected as lax *ö, never tense *öö.
- In Permic, at least in open syllables *i-ä is lowered to a secondary *e (which in Komi furthermore may yield /o/). No similar lowering applies to *ü in at least *küsä > *kɨz ‘thick’. OTOH we do have *d₂ümä > *ĺem ‘glue’?
- I was originally also planning on mentioned some oddities involving *ü in Mansi here, but these have since the inception of this post gotten expanded into a reanalysis that I think solves them.
This could suggest that *ü was originally “one step behind” PU *i and *u, i.e. some sort of a tenser vowel. An extended system of close vowels as assumed by Mikola, Katz, etc. would have room for such a thing: assume a PU contrast between tense *i *ü, and lax *ɪ *ʊ? But perhaps not. I can think of other explanations for these issues as well. Going into extended detail is not possible here, but in brief:
- The alternate reconstruction of Khanty *ee, *oo as lax open *ä, *a (as still reflected in the Surgut dialect) may provide a clue to understanding this asymmetry. Assuming in pre-Khanty a symmetrical situation with *e, *ö, *o for PU *i, *ü, *u, we can expect only *e and *o to continue downwards to *ä and *a. For one, roundedness contrasts in open vowels are highly marked; for two, “pressure” from secondary lax *i and *u (from PU *e and *o in roots of the shape *CVCə, and producing lax *e *o in Proto-Khanty) would have provided a motivation for the lowering of these, while there was no secondary lax *ü.
- The early Permic vowel system, before the lowering of *i, likely had a mid front gap, brought about by the early retraction of PU *e (ultimately to Udmurt /u/, Komi /o/). In this context, a split of *i to *i, *e makes a decent amount of sense. For ‘glue’ I suspect a fronting *ɨ > *i before this, brought about by the palatalized onset consonant.
And so for now, I have no satisfactory account of the hows and whys of the development of *ü in Samoyedic.
 Strangely, I have found no indication of anyone having previously proposed this etymology. It is entirely unproblematic both semantically and phonetically. A small hurdle might have been that Janhunen (1977) reconstructs this as *tajkå instead, but his choice of *a and not *ə seems arbitrary. The Northern Samoyedic reflexes are somewhat irregular, while Southern Samoyedic does not distinguish *ə from *a.
 Also allegedly reflected in Mordvinic *pižə ‘green’. I suspect that these words derive from some descendant of Indo-Iranian *wiša ‘poison; green’. I have also entertained the idea that PSmy *pətä might be a metathesized reflex of PU *säppä ‘gall’ (the predicted reflex of this in Samoyedic would be *täpä or *täpə), but the II connection seems more promising than a random entire-syllable metathesis.
 It is only after the introduction of reduction that the trajectories of front *ɪ and back *ʊ seem likely to diverge. The shift *u > †ʊ > /ʌ/ in most dialects of English is one good example; the shift *i > *ɪ > *ë > /ɑ/ in large parts of the Samic languages is another. But we still also find *i > /ɪ/ in English, and *u > *ʊ > /o/ in Samic. If there are examples where one close cardinal vowel is reduced while another remains, I have not seen it.
 E.g. *jupta- ‘to say’ > *jəptə- ‘to count’, *kod₂ka > *kåjkə ‘spirit’, *mëksa > *mïtə ‘liver’, *peksä- > *petə- ‘to beat’. — A possible exception to this pattern though seem to be consonant clusters ending in a labial consonant: e.g. *äjmä > *äjmä ‘needle’, *kompa > *kåmpå ‘wave’, *ojwa > *åjwå ‘head’, *päjwä > *päjwä ‘sun’, and the above-mentioned *śilmä > *səjmä ‘eye’. Speculating a bit, I wonder if this could be connected to the fact that labial consonants are in Uralic common in the word-initial position and in derivational suffixes, yet word-medial *p is quite rare. Might these words have been in Proto-Uralic not indivisible roots, but instead derivatives and/or compounds? This would allow e.g. comparing the first syllable of *äjmä with PIE *h₂aḱ- ‘sharp’.
 Janhunen, Juha (2013). “Personal pronouns in core Altaic“. In: Robbeets, Martine; Cuyckens, Hubert (eds.) Shared Grammaticalization. With special focus on the Transeurasian languages.
 This reconstruction might be preferrable to traditional *lexə-: it fits Finnic *löö-, Mari *lüje- and Hungarian lő equally well, while Komi /lɨj-/ explicitly suggests a close labial vowel *ü or *u.