Ranko Matasović, in a recent paper “Substratum words in Balto-Slavic“:
Balto-Slavic also has a number of verbal roots which do not appear to have any cognates elsewhere. (…)
• BSl. *pel-/ *pāl- ‘burn’ > PSl. *paliti ‘burn’
I will take his word on the nonexistence of clear Indo-European cognates. However, we can find a near-identical root right next door in West Uralic (= Samic-Finnic-Mordvinic): *pala- ‘to burn’. This seems like a much clearer point of comparison than Matasović’s proposal of metathesis from PIE √leh₂p- ‘to shine’.
A traditional further comparison within Uralic has been with with Ugric *pad₂ɜ- ‘to freeze’. I’ve never found this compelling. The semantics display a “thermal inversion”, and phonologically this only works by recourse to the dubious PU *ľ, and even then only halfway: in Khanty we’d expect then *Ľ, not *j. I’m more inclined to accept instead the recent connection in Aikio ’16 with the long-known word-family of *pala ‘bit’, more specifically with verbal reflexes in Ugric and Samoyedic meaning ‘to devour’.
Originally I was planning on simply quoting the Uralic material and concluding with “∎”, but no, this does not yet add up to a trivial etymology. For one, even though the narrower distribution clearly suggests West Uralic → Slavic, we could still consider also the opposite direction of loaning (at the cost of abandoning the East Uralic cognates, geographically too far off to be of Slavic or even Balto-Slavic origin; but also at the benefit of dispensing with the semantic shift from ‘devour’). For two, the correspondence Uralic *a ~ Slavic *a poses some difficulty. These are identical graphemes — but before the Great Common Slavic vowel shift, the latter “*a” would have been phonologically a long vowel *ā. [1] Could *a on the Uralic side, not originally subject to a length distinction, been substituted as long *ā > CSl. *a, instead of short *a > CSl. *o? Possible, perhaps, but no such phenomenon surfaces in any of the (rather few) known old Uralic loans into Germanic and Baltic. Alternately, could this be a loan late enough to have skipped the CSl. vowel shift altogether? Again, maybe this is possible. But we clearly end up with some uncertainties in how this supposed loan could have been routed.
For three, the WU root alternates with an “ablaut variant” *pol-(t)ta- ‘to burn (tr.)’, which has never been properly explained. Under current knowledge, we could maybe derive the Samic and Mordvinic variants (*poaltē-, *pultə-) from earlier *palə-ta- ~ *palə-tta-, though Finnic still remains problematic. [2] The existence of comparanda in Slavic opens some new options, though. Some kind of back-loaning is one possibility; for another one, since I am not 100% convinced that this is a U → Sl loan and not the opposite, maybe we could derive the Uralic variants from actual IE ablaut variants, such as an earlier full grade *pōl- versus an otherwise lost zero grade *pal- (from earlier *poh₃l- ~ *pᵊh₃l-)?
Later on in the paper, Matasović also gives a list of various voiced/voiceless doublets, mostly from Baltic. He then adds a strange comment: “In some cases, words showing this alternation may be Uralic loanwords, or they may reflect the pronunciation of originally Baltic words by speakers of Uralic, who underwent language shift.” This does not seem to be combined with any attempt to find Uralic equivalents though… and in many cases such a search would be doomed anyway. At minimum, doublets with word-initial consonant clusters (a bit over half of the cases, e.g. Latvian sniekt ~ sniegt ‘to give’; Lith. klusnus ~ glusnus ‘obedient’) would be clearly alien to Uralic phonology. The only case in his list that I could possibly see as connected to any actual Uralic words is Lithuanian viskėti ~ vizgėti ‘to swing’, which has some similarity with Finnic *viskat- ‘to throw, cast’ (but maybe not enough to actually matter).
I don’t want to harp on Matasović in particular. But this regardless strikes me as a part of a wider disconnect between IE and Uralic studies. The oversights here — the false negative of *paliti being supposedly isolated, and the (weak) false positives of words like klusnus being called possibly Uralic — both fit into a pattern where Uralic gets unwarrantedly treated as lexical terra incognita, despite extensive research to the contrary; much of it in readily accessible languages like German and English, even.
Plenty of Uralic-IE lexical comparisons have of course been compiled over the times… by Nostraticists and Indo-Uralicists. Skepticism on macro-comparison hypotheses like these should not be taken as a reason to neglect the raw data used, though: by now many cases have been shown or at least proposed to be loans from IE to U instead. I would expect close analysis to also reveal some number of cases better explainable as U to IE loans, too, if we would only pause to at least consider the possibility. [3]
[1] This type of error is committed every so often in IE/U loanword studies, where numerous traditional transcription schemes clash. Also worth mentioning is at least the similar graphemic identity but phonological non-identity of PU *a (an open back vowel, [ɑ ~ ɒ]) and PII *a (a non-close central vowel, [a ~ ə]); which explains e.g. why PII *ćata ‘100’ gets loaned as PU *śëta (with *ë, a mid non-front vowel, [ʌ ~ ɤ]) rather than *śata.
[2] I’ve sometimes wondered if the stem type shift *a-ə > *o-a could have under some conditions, such as after *p, extended to Finnic as well. However, by this approach it would be mysterious why we end up with Fi. polttaa and not a contraction verb polata : polaa- (*polat-). Estonian põlema ~ Votic põlõa ‘to burn (intr.)’ also seem to evidence instead a base root *polë- < *polə-. One other solution would be to suppose instead *palə- > *poolë- (just as in *pälä > *palə > *pooli ‘side, half’), followed by suffixation to transitive *pool-tta-; which might have been then shortened to *poltta-, due to a general ban on CVVCC syllables in Proto-Finnic times. After this Estonian and Votic might have back-derived the stem *polë- from this…
[3] For one other possible example, cf. the case of Uralic *pisə- ‘to put (in)’ ~ Balto-Slavic *p(e)is- ‘to push, to fuck’, briefly discussed in this blog’s comments earlier.
Freelance reconstruction 
Skok considers *paliti a causative, which doesn’t explain the long vowel entirely, but opens way for some creative laryngeal/analogical apophony explanations. The same goes for the possibility of it being an iterative formation.
He also cites Čakavian spolīti “anbrennen” and OCS poleti “uri, ardere”.
Other related words to keep in mind are *pepelъ “ash” with Lith. pelenai “id” and various IE “dust” words traditionally associated (see Lat. pulvis), and Lith. pelenas “fire-place”.
Frostbite?
Surely gt is pronounced [kt]? If so, can’t the doublets in whatever inflected forms be due to analogy?
Nitpick:
A burning object IS devoured/bitten by flames; it would be weird to view it as devouring other objects (by spreading the fire).
Therefore, if a word steps in meaning from eating to burning, I would generally expect some kind of derivational process to account for the change to passive meaning.
Why would Western Uralic not do that?
I’d blame how Uralic languages tend towards underived basic intransitive verbs vs. derived transitive, reflexive etc. verbs. In this case, too, we see the introduction of a derived transitive *polə-Ta-. Note also that western *pala- is in its semantics more like middle voice ‘to be burning’; ‘fire burns’ than passive ‘to be on fire, burnt’; ‘wood burns’.
Aikio proposes a slightly different route, through reanalysis: “The meaning ‘burn’ in West Uralic derives from a metaphorical expression “fire eats”, which has parallels in Ob-Ugric and Samoyed languages.” Surely possible, but I am unsure if necessary.
I wonder about opposite issue, though: why do the East Uralic verbs pointing to *pala- ‘to eat’ not contain a verbalizing suffix of any kind, even though they seem to be derived from the noun *pala ‘bit’?