Thee, thou, and thine (or thy) are Early Modern English second person singular pronouns. Thou is the subject form (nominative), thee is the object form, and thy/thine is the possessive form.
Before they all merged into the catch-all form you, English second person pronouns distinguished between nominative and objective, as well as between singular and plural (or formal):
thou - singular informal, subject (Thou art here. = You are here.)
thee - singular informal, object (He gave it to thee.)
ye - plural or formal, subject
you - plural or formal, object
Interestingly, when the first English translations of the Bible were being made, the informal thee and thou were used specifically in reference to God to indicate an approachable, familiar God, but as the language changed this paradoxically brought thee and thou to sound more formal to the modern English speaker.
Transitive swear and Concessive Subjunctives
I don’t know why you think swear is intransitive; it is obviously used transitively here since it is love that would be sworn. If she can swear love, then swear has to be transitive: it has an object.
But had you checked a dictionary, you would have found numerous transitive senses for swear. Indeed, the OED gives no fewer than a dozen transitive senses and subsenses for this verb. For just as you can swear fidelity, so too can you also swear love.
Now that we have that out of the way, what remains is nothing fancier than commonplace sentences like these:
Do it and you’re a dead man.
Give it up and we’ll go home.
Stop over and we’ll talk about it.
Be kind to strangers and strangers will be kind in return.
This type of construction includes “be but sworn my love”. If Juliet swears his love, she attests to it as by promise or oath — upon which circumstance he pledges to renounce his own family.
Technically speaking, this is a concessive subjunctive construction, the same thing we find in frozen refrains and familiar aphorisms nestled in sentences like these:
Be it ever so humble, there’s no place like home.
Give him an inch and he’ll take a mile.
Come rain or come shine, the cows will still need a milking.
Take what you will of me, I will never complain.
As you see, the verb in concessive subjunctive clauses likes these takes a simple infinitive as far as its morphological inflection (or lack thereof) goes.
Shakespeare uses concessive subjunctive constructions quite frequently throughout his writings. In the selfsame play, we later in Act 3 Scene 1 find the following famous example from Mercutio perishing by the sword:
No, ’tis not so deep as a well, nor so wide as a
church-door; but ’tis enough, ’twill serve: ask for
me to-morrow, and you shall find me a grave man.
Although Latin teachers go to some effort explaining Latin subjunctive inflections in concessive clauses, even in uninflected English this sort of thing still happens — at least as far as having concessive clauses goes.
These are really nothing fancy, as the first set of examples in the earlier section prove. Concessive clauses still happen in regular English outside of fossilized phrases like “Be that as it may”, and when they do, they take a bare infinitive in English.
Last and probably least, I believe I have by now also answered by way of earlier demonstration your remaining question as to but’s function in the love-sworn citation.
Best Answer
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
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):
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:
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).