Has the pronunciation of Greek changed since the Byzantine Empire’s collapse?

Since 1453? Hm.

It’s hard to pin this down, because Greek at the time was a whole bunch of dialects, whose pronunciation we don’t have a good handle on historically—but which was likely stable. (There aren’t any surprises in the Renaissance Latin alphabet transcriptions of Cretan for example.) For that matter, Standard Modern Greek did not coalesce until the Modern Greek state was established.

The one area where there has been recent pronunciation change in Standard Modern Greek is in the extent of prenasalisation in the compounds <μπ, ντ, γκ>. There is an isogloss separating dialects which pronounce them as [mb, nd, ŋɡ] (e.g. Cyprus) from those pronouncing them as [b, d, ɡ] (e.g. Crete). Within Standard Greek, the shift has been from the former to the latter, and the shift happens with people who are now in their 50s.

The question adds whether Turkish has had much of an impact on pronunciation. As far as I know, noone’s claimed it or expects it: Turkish settlement in the Balkans does not seem to have been substantial enough to have had much impact. The Greek spoken in Cappadocia would presumably have had substantial phonetic impact from Turkish; but Cappadocian was under immense linguistic pressure from Turkish.

Turkish has somewhat complicated the phonology of Cypriot Greek, but not its phonetics. Cypriot Greek already had geminated stops realised as aspirates: potʰe ‘never’, tʰofis ‘Chris’, and it already realised palatalised /s, x/ as [ʃ]: xiros > ʃiros ‘pig’, skillos > ʃtʃilːos > ʃilːos ‘dog’. Turkish stops and <ş> took on the same pronunciations, they just turned up in places where they would be rare to non-existent in Greek: at the start of words (for stops), and at the end of words (for ʃ): kʰele < Turkish kele ‘head’, paʃ < Turkish baş ‘chief’. The kʰ and ʃ were already in the dialect, just not in those positions before Turkish contact.

(Hence me staring at εξίκκον σου on Facebook, trying to work out what Ancient Greek phrase lurked behind ‘exikkon to thee’ = “it’s not worth the bother”. In fact, this is just /eksikkossu/ (with folk etymology reinterpreting –ossu as the more hellenic –on sou), and [eksikʰosːu] is merely the Turkish eksik olsun “may it be missing”.)

But other than that, no instances of [ɯ] or [ø] (outside of Cappadocia), and no particularly telltale Turkish intonation (outside of Cappadocia). Greek mangles Turkish loanwords to fit its phonetics, and much of the time its morphology; so cacık [dʒadʒɯk] > tzatziki, Karagöz [karagøz] > Karagiozis /karaɡjozis/.