I haven't found a source that discusses this exact rhyme pair yet, so what follows is just speculation that should be read skeptically.

The vowel in feast

It seems possible to me that it might have been pronounced as [fɛst] or [fest], with a short vowel. (Most varieties of English do not have a distinction between the qualities [ɛ] and [e] for short vowels, even when these qualities are distinguished for long vowels, so it's hard to tell which is the more accurate transcription.)

The modern English pronunciation of feast, /fiːst/, has a "long e" sound which would correspond to the Middle English long vowel [ɛː]. The corresponding Early Modern English pronunciation is often reconstructed as [eː], which is intermediate in quality between [ɛː] and [iː].

This vowel was spelled ea, ee, and e in Middle English, and the Oxford English Dictionary records each of these spelling variants for the vowel in feast. (The letter "ē", mentioned in a comment, was not used as a symbol for a long vowel in Middle English; the use of the macron diacritic to represent a long vowel is a modern convention.)

Why the vowel is long, and how a short variant might exist

I think the most likely reason that a pronunciation of feast with a long vowel existed is because some varieties of French had a sound change that lengthened vowel sounds before /s/ (somewhat similar to the British English sound change lengthening /a/ to /ɑː/ in words like last, but the French change applied to more vowels than just a). So Latin [fɛsta], with a short vowel, may have developed to [fɛːstə] in French before being taken into Middle English as [fɛːst(ə)].

However, it may be possible that a form with a short vowel was also taken into English, either before the lengthening sound change occurred in French, or from a variety of French that did not undergo vowel lengthening. I'm unfortunately not sure of the exact location in time and space of the French vowel-lengthening sound change: the Wikipedia article "Phonological history of French" lists "/a/ develops allophone [ɑ] before /s/" as a c. 1100 sound change but "lengthening of the preceding vowel" as a c. 1250–1300 sound change (treated as part of the s-loss process which was obviously not complete at the time that feast entered English).

The Oxford English Dictionary's first citation for feast in English (spelled feste) is dated c1225, so I think it's possible that the French lengthening sound change was still incomplete around the time that the word was entering English. There are words from French that are currently pronounced as monosyllables with a short vowel before st, such as jest, test, crest, pest, vest. Going against my hypothesis, though, these seem to mostly be more recently attested than the words currently pronounced with a long vowel (feast, beast), so I'm not completely sure what causes the divergence.

In the French spoken in Shakespeare's time, words like feste (French) were pronounced without [s]; i.e. as [fɛːtə] (see A Dictionarie of the French and English Tongues, 1611, compiled by Randle Cotgrave, printed by Adam Islip, page 2). So Shakespeare's usage is presumably not based directly on the pronunciation of French during his lifetime.

I think it's important to note that we also see Shakespeare rhyme French words in -aste with words in -ast: there are at least two examples in Romeo and Juliet of haste/last ("O, let us hence; I stand on sudden haste. / Wisely and slow; they stumble that run fast." and "Therefore use none: let Romeo hence in haste / Else, when he's found, that hour is his last.").

David Crystal, who is well known for studying Shakespeare's pronunciation, seems to think that Shakespeare had a short vowel in waste and taste ("Sounding Out Shakespeare: Sonnet Rhymes in Original Pronunciation", page 298). Unfortunately, I don't own any of Crystal's books, so I can't give a more detailed description.

Near rhyme hypothesis

An alternative interpretation of this data could be that Shakespeare simply resorted to near rhymes between short and long vowels in some cases. In that case, the pair feast/guest might be something like [fɛːst]/[gɛst], [feːst]/[gest], or [feːst]/[gɛst].

One piece of evidence that I think is moderately strong for this is that Shakespeare also rhymes east—which is not from French, but is a native English word where the ea is from an originally long diphthong—with short -est words. This could indicate that Shakespeare's accent had shortening of e in this word, but that's a more complicated scenario (because it assume divergent developments of the Old English vowel in Shakespeare's accent and modern English accents) than assuming that Shakespeare had a long vowel in this word. Examples:

Sonnet 132:

And truly not the morning sun of heaven 1840
Better becomes the grey cheeks of the east,
Nor that full star that ushers in the even
Doth half that glory to the sober west,
As those two mourning eyes become thy face:

The Passionate Pilgrim:

Lord, how mine eyes throw gazes to the east!
My heart doth charge the watch; the morning rise 195
Doth cite each moving sense from idle rest.
Not daring trust the office of mine eyes,
While Philomela sits and sings, I sit and mark,
And wish her lays were tuned like the lark;

The vowel in guest

Guest is a word with a fairly irregular development, mainly in terms of the initial consonant. The Oxford English Dictionary points out that the Old English form giest/gist/gyst/gæst should have developed to a present-day form with the palatal glide /j/; the velar /g/ could be due to Old Norse influence. Neither source is expected to yield a form with a long vowel in Modern English; however, the Oxford English Dictionary records a few Middle English spellings with digraphs for the vowel: geest and geast(e).

I would say the evidence for a possible long vowel in this word seems pretty slim, although it isn't entirely impossible (yeast shows an anomalously long vowel).

Also, given that Shakespeare also rhymes feast with best, you would need to assume lengthening in best as well to support the hypothesis that Shakespeare was using only perfect rhymes and had a long vowel in feast. So I don't think this rhyme pair strongly supports the idea that Shakespeare pronounced guest with a long vowel, but the idea should at least be considered.