Yes. Homer's works were written in Homeric Greek, an ancient dialect with elements of Ionic/Aeolian Greek. Alexander may have known this dialect through his studies, but he would have regularly spoken the common dialect (koine dialect), an offshoot of Attic Greek. This was somewhat developed by his father, Phillip II of Macedon and spread by Alexander during his conquests (hence its other name, Alexandrian Greek).

Considering as well most playwrights were writing in Attic Greek and referenced Homer frequently, it's safe to say that there were transliterations so it would be easier to read for the literate. The major source of modern scholarship on the Iliad, the Venetus A, only dates back to the 10th century CE, but itself offers clues that it is derived from the works of Greeks who wrote in koine dialect. However, many of the original works are now lost.

Side note: I believe Arrian, Alexander's biographer, wrote in Attic Greek.

I'm going to say that the answer is no, and that /u/pipsdontsqueak is making an argument from silence that is open to serious objection.

It is true that Homeric Greek, being a combination of dialects and emerging aroudn the 8th century, was not readily understandable to later speakers, certainly to Koine speakers. But, neither are Shakespeare's works readily understandable to us.

So, yes, Greek language evolved and there was a linguistic gap. But, there's no evidence that 'transliterated' (that's not really the right word) or 'updated' texts existed. That's a huge gap in the documentary evidence and I don't think it's safe to 'fill it in' by assumption.

What we do have are two pieces of evidence that would argue in a different direction. 1) We have a reasonable number of texts that outline grammar and rhetoric, and included within them are the steps a student ought to take when analysing a poetic text or similar. One of my favourite examples is Dionysius Thrax's Art of Philology, which outlines the steps of 'exegesis' involving:

1. Clarification of a word's meaning
2. grammatical and rhetorical analysis
3. metrical evaluation and style criticism
4. analysis of the historical realities of the text.

These are the 4 substages of a larger 4-step process. Anyway, the point is that a student of Greek literature would be expected to note odd and archaic words, give an account of their etymology and meaning, place them in context, and discuss how they contributed to Homer's poetics. This suggests that rather than 'update' Homeric works, students were trained to read them as is.

2. My second piece of evidence are the scholia, essentially commentary notes written on copies of Homer's works. These are written in the margins, between the lines, and at the start and ends of pages or sections. They are the equivalent to annotated versions. Again, this suggests that Homeric (and other) texts were not 'updated' but merely annotated.

All in all, I would argue that the evidence points in the direction of providing aids for students of Homer and other more ancient literary works, rather than providing 'modern versions' for later Greek readers. Greek rhetorical education was aimed at producing competent readers of their own antique works, so that even Koine speakers could, in late Antiquity, read, write, and analyse Attic and Homeric Greek at a high level. The absence of any particular manuscript evidence for 'updated' versions weighs heavily against it, while both the evidence of literary critical practice, and the scholia traditions of commentary, argue that such texts were made 'understandable' by ancient versions of Student texts and Cliff notes.

Followup question: were there works not updated because it was thought that the updates took away something from the story? For example, many people read the original Shakespeare and his plays are still performed as written, would Greeks do so with one of their plays?