Anyone with even a passing interest in poetry (like me!) will have seen at least a few examples. Three off the top of my head are "Death, be not proud" by John Donne (1633; rhymes eternally/die), "The Tyger" by William Blake (1794; rhymes symmetry/eye), and "Sonnet—To Science" by Edgar Allan Poe (1829; rhymes eyes/realities).
Which words changed pronunciation, over what period of time did it occur, and how can we tell? (I assume Poe uses it as a deliberate archaism, but does Blake as well?)
Historical phonology is not my area of expertise, so apologies if this answer is not as comprehensive as you might expect from a flaired user, but as a theatre lover I do have a passing interest in Original Pronunciation (English as it was pronounced in Shakespeare's day), so hopefully I can get the ball rolling (with the view that someone more knowledgeable would expound).
Almost certainly in Donne's time, "eye" was pronounced similarly to the present-day pronunciation, while "eternally" would have been closer to "eternal-lie". The Oxford Dictionary of Original Shakespearean Pronunciation (OUP 2016; edited by David Crystal, an expert on the historical phonology of Early Modern English), which compiles the reconstructed pronunciations of words c. 1600, lists the following entries:
eye - ǝɪ
eternally - ɪˈtɐ:ɹnɑˌlǝɪ
You don't have to know the International Phonetic Alphabet (IPA) to observe that they have the same ending! Though note that the present-day pronunciation is very slightly different: aɪ rather than ǝɪ for eye (I suppose a very rough and ready description would be that ǝɪ would be closer to how someone with a strong modern Irish accent would pronounce "eye".)
As for the reconstruction process of Elizabethan pronunciation, Crystal has this to say in the foreword to his dictionary:
A reconstruction exercise is by no means straightforward because the textual evidence is often difficult to interpret. A distinctive spelling may genuinely indicate how a word was pronounced, or it may be a typesetter’s error. Words at the ends of lines may point to a genuine rhyme (as in a sonnet) or may have a fortuitous connection... The possibility of eye-rhymes [words that end with the same letters but are pronounced differently, like Sean Bean] also needs to be considered.
Notwithstanding, Crystal observes that "there are a sufficient number of clear cases of spellings, rhymes, puns, and comments to warrant a reconstruction of the English sound system of Shakespeare’s day". For a more comprehensive review of the methodology of phonological reconstruction, he recommends Eric Dobson's English Pronunciation 1500–1700 (2nd edn, Clarendon Press 1968), which unfortunately I don't have access to. As such I'm afraid I can't answer the second part of your question, that being when the pronunciation changed to its current form. Hopefully someone more knowledgeable will chip in with a fuller answer.