Not too distantly, from what we can know. Much of our reconstruction of Elizabethan pronunciation is to be quite frank based on Shakespearean puns, rhymes and meter (along with the remainder of the dramatic and poetical corpus otherwise). We can say at least, for certain, how many syllables a word possesses, from its metrical use. And by comparing rhymed and punned words (i.e., cases in which a word is meant to suggest an alternate meaning via homophony), we assemble a picture of their pronunciation.

From Shakespeare (who uses the name in blank verse, in Richard III and Henry VIII), we know that in colloquial Early Modern English, "Elizabeth" is a four syllable word with emphasis on the second and fourth syllables, just as in our contemporary English. From Richard Corbett, we know the name can be satisfactorily rhymed with "heath", suggesting at least here, the final vowel is not meant to be a schwa (as it often would be in modern colloquial pronunciation) but instead closer to /e̞/ (a vowel the modern American reader may approximate by considering a sound intermediate between the vowel in General American 'bell' and the vowel in General American 'bale'). Ben Jonson corroborates this reading by rhyming the final syllable with "death". A stressed /ɪ/ such as we see suggesting itself in the second syllable is among the least suspicious of all vowels in Early Modern English. Were the name a conservative survival from Old English or colloquial Early Middle English, there might be room for consideration of an Early Middle English /i/ becoming /aɪ/ amidst the Great Vowel Shift before reducing itself to /ɪ/. But I would argue that "Elizabeth" is not so old nor so commonplace a name as will allow for such possibilities in the late medieval context to impose themselves on the Early Modern.

In short, there are no significant sound changes which especially much apply to this name's pronunciation. But we may infer from its usage in the Elizabethan period that the final syllable's reduction to a schwa (as is common in contemporaneity) was not convention, and we may suppose that that most durable and reliable of Modern English vowels, /ɪ/, was unaffected. There is room for variation in the pronunciation of 'a', and this could well be reduced to a schwa. But it is difficult to say, and might very well simply be circumstantial, as vowel reduction frequently is today.