Qualifications
Rare book dealers are usually self-employed, so there's technically no degree required. However, you do need to be really well-versed in your subject. If you have zero literary knowledge, there's a very good chance you won't make it far in the rare books business. Sorry to be the bearers of such bad and very obvious news.
When a customer walks up to you at the book fair and asks if you know where she can lay her hands on a first edition of The War of the Worlds, you'd better know she's talking about the novel by H.G. Wells and not the movie starring Tom Cruise. To avoid such embarrassments, you ought to get a degree in English or creative writing, and maybe even throw in a grad degree in comparative literature. Your continuous goal is to be the most knowledgeable dealer at the rare books fair.
Aside from knowing your stuff when it comes to literature, you'll also have to be an appraisal expert. There aren't a whole lot of degree programs for this; it's one of those things where you'll need some hands-on experience. Apprenticing yourself to an experienced rare book dealer is a great way to learn.
If you're more of a lone wolf, you should become a regular at book fairs. You should also become really well-acquainted with websites like Biblio, which will let you see the prices that books are going for these days.