Except for law (which an accredited undergraduate degree is legally required in most states), graduate schools don’t even require a BA to get in, but it certainly helps. Getting accept to a graduate program requires, typically, an undergraduate degree, and a transcript that shows you can do the work required, and it helps if you can convince them you know what you are doing. Fro example, if you come to an interview and can switch between Hebrew, Arabic, Aramaic (a.k.a. Syriac), Farsi and Turkish — they will be very impressed if you are applying for a program in middle eastern languages. Want ancient near eastern studies: write your application letter in Akkadian.
But for normal people, they look at your undergraduate transcript, references, and anything else you can use to “sell” yourself, and a BTL from a yeshiva is not especially prestigious. It does mean you are qualified to apply without asking for an “exception”, but if all your transcript has is evidence of studying in a yeshiva you may not be in a great position.
And of course, many schools are very happy to admit people who pay full tuition, and even more so if your family wants to endow something.