Swedish doesn't really do double vowels, and when you make a neutral noun definite, you should only end up with one vowel before the t. So why is "the tea" "teet"? Is this just a weird exception or is there some rule here?
Swedish does in fact do double vowels, usually with the letter "e" as you've seen. There are several words that do it, and not just in the bestämd form. The reason for keeping the double vowel is mostly pronunciation. If you were to write 'tet' instead of 'teet,' it would be pronounced with a short vowel instead of the lilting, "tea-et" which it ought to be- hence the double vowel. There are several words with double vowels besides, by the way, such as:
Most nouns, like you said, don't add the extra "e" in the bestämd form, for the reason that it isn't pronounced with the "lilting" accent used for "teet," wherein both e's are pronounced. Like:
öde (fate) -- ödet
yrke (occupation/profession) -- yrket
öre (hundreth of a krona/penny) -- öret
tet could only be pronounced as [te:t], with a long vowel (cf. vet 'know[s]'). teet is pronounced with two vowels, [te:ət], which is why it's spelled with two e's.
Maybe 'tet ' means something too. In Flanders that is used als informal for breast, maybe in Swedish too?