Do Aeolic and Ancient Greek have other examples of τ/π in addition to the pair πέντε/πέμπε?
Andrew L. Sihler's New Comparative Grammar of Greek and Latin's treatment of the phenomenon additionally mentions Lesbian πέσυρες, Homeric πίσυρες, Boeotian πέτταρες = Attic τέσσερες 'four', and Lesbian πήλυι = Attic τηλοῦ (Homeric τῆλε) 'from afar' by way of example.
All reflect Proto-Indo-European *kʷ, with the general rule being that it became τ before a front vowel and π or κ elsewhere, except in some basic words in Aeolic where it became π even before a front vowel; compare e.g. Attic πεμπάζω 'count on the fingers', where the same *kʷ became π before the α. (μ rather than ν is due to assimilation to π.)
Note that e.g. τίς < PIE *kʷis and τε < PIE *-kʷe are the forms in Aeolic as well, rather than ˣπίς and ˣπε. The exception is not universal.
This phenomenon isn't confined to the voiceless stops τ/π, but also involves their voiced and aspirated counterparts, δ/β and θ/φ. So there are pairs like Δελφοί / Βελφοί "Delphi", θήρ / φήρ "wild animal", Φετταλός / Θετταλός "Thessalian".
The simplest way of describing things is to say that Proto-Indo-European labiovelars became dentals before front vowels and labials elsewhere, except in Aeolic where they became labials across the board. (There's also a separate conditioned change to velars which can be disregarded here.) But there are exceptions to both parts of the rule: τίς, τε in Aeolic as Cairnarvon mentions, but also e.g. βίος in all dialects from *gʷ- despite the front vowel. (Actually "before front vowel" is an oversimplification as the treatment before i seems to have differed from that before e in some respects.)
In more phonetic/phonological detail, the process may have gone something like this:
Labiovelars develop an allophone of some kind, presumably a palatalized velar, in the environment "before front vowel". (There's evidence for this from Arcadian, which in one inscription uses a special letter И for labiovelars in this environment.) However, this does not happen in Aeolic, or at least not to the same extent.
There is then a pan-Greek unconditioned change "labiovelar > labial". But this does not affect the palatalized velar allophone from (1), which is no longer a labiovelar.
The palatalized velars -- now no longer allophones, but phonemes with restricted distribution -- merge into the existing dental series (though in Arcado-Cyprian they sometimes became sibilants instead of stops).
(This might explain the Aeolic exceptions τίς, τε if their frequency somehow led either to their being more palatalized than other such words in Aeolic or, maybe more likely, to their being perceived as phonologically different enough from other labiovelars that they were spared by the change in (2).)