Caveat: I am not a Shakespeare scholar or any kind of expert on the language of this period. I feel fairly confident about my explanation in this post of the use of "thee", but not totally confident about my explanation of the use of "dislike".
The use of thee instead of thou: an old, now-archaic construction with the verb "(dis)like"
You are certainly correct about "thee" being in a non-nominative case, rather than the nominative case form "thou" that we would expect for the subject of a sentence.
However, the meaning does seem to be "if you dislike either" or "if either displeases you". We can tell this from the context, and this interpretation is also supported by the Oxford English Dictionary's entry on "like", which says that in the past, it could be used
. trans. To please, to be pleasing or agreeable to, to suit (someone). Now arch.
Originally with dative. In early use also intr. with to, of, or till.
So, the usage of "like" has changed: we used to be able to say things like "it liketh me" or "me liketh it" to mean "I like it." Sebastian Redl left a comment pointing out that Shakespeare does use the word "like" in the modern way in many places.
But I think there are also other examples of the old-fashioned construction being used in Shakespeare's work. I searched "Open Source Shakespeare" and found these (not a comprehensive list):
Claudius. It likes us well; /
And at our more consider'd time we'll read, /
Answer, and think upon this business. (Hamlet, II, 2, 1173-75)
Hamlet. This likes me well. These foils have all a length? (Hamlet, V, 2, 3907)
Sir Thomas Erpingham. Not so, my liege: this lodging likes me better, / Since I may say 'Now lie I like a king.' (Henry V, IV, 1, 1859-60)
Cassio. I'll do't; but it dislikes me. (Othello, II, 3, 1178)
My guess is that it may have been a construction that was on its way out already in Shakespeare's time (like the use of "thou" and "thee"), so he used it inconsistently, and more often in passages with other somewhat old-fashioned language. I actually can't find any use of "liketh" or "disliketh" in Shakespeare: these forms may have already been relatively archaic in his time. I believe the use of -(e)th instead of -(e)s at this point was more common for the auxiliaries "doth" and "hath" than for ordinary verbs. (For comparison, Shakespeare did use the word "wanteth", but not very frequently: Open Source Shakespeare turns up only six examples of wanteth compared to around 33 examples of wants as a verb. It seems possible that some of Shakespeare's use of -eth on non-auxiliary verbs was often motivated by considerations of scansion, as mentioned in Tim Lymington's answer and Peter Shor's comment below it.)
This is actually, I believe, a sort of construction that exists across the Germanic languages, and there are various ways to interpret it. ("Methinks" is another example.) The OED entry suggests that it is a transitive verb taking a dative object; however, in languages like Icelandic, similar verbs are analyzed by some linguists as having what is called a "quirky subject" that is not in the nominative case, but that nevertheless plays the syntactic role of a subject (although it does not trigger agreement on the verb: the verb agrees with the noun phrase in the nominative case). The "quirky subject" analysis is fairly complicated, and also disputed, so I don't think I can explain it any better than that.
Actually, identifying the subject is a bit immaterial. Whatever we analyze as being the syntactic subject, we can infer that the noun phrase "either" in this clause must be in the nominative case, and so by all accounts the verb should therefore agree with "either", which is third-person singular.
The use of dislike instead of dislikes: I think it's subjunctive
The use of "dislike" rather than "disliketh" or "dislikes" seems to me like it must be an example of the present subjunctive, which was formerly possible in this kind of "present-conditional" clause. E.g. it's analoguous to the archaic use of expressions like "if it be..." rather than "if it is...." More examples of this construction are mentioned in the answer to the following question: How to interpret “if it be” grammatically?
From Shakespeare specifically, here's one example that I think seems similar:
Helena. [...] I am undone: there is no living, none, / 85
If Bertram be away. (All's Well That Ends Well, I, 1, 85-86)
So the overall structure of "if either thee dislike" would be "[nominative 3rd-person singular NP] [non-nominative 2nd-person singular NP] ["present subjunctive" verb]". An updated version with different words but a similar kind of word order and verb-agreement pattern would be "if either you displease" (= "if either displeases you", in a version with entirely modern grammar).