You would need a public performance license. If this was playing music CDs, you would get in contact with ASCAP / BMI for the license. But I'm not sure what kind of central clearinghouse exists for pubic performance licenses for DVDs. You may have to contact the legal departments of all the rights holders of the particular DVDs you want to show. I'm sure this isn't what you wanted to hear.
EDIT Just look at this link:
Motion Picture Association of America The magic words for Google were "video public performance license". I'd guess the MPAA would "front" the major studios and that Criterion would "front" most of the art films you describe. Both have toll-free numbers at that link.
EDIT EDIT I'll bet MPAA is going to be much easier to work with than your local health department in getting a license to serve hot food.
Not to mention the local alcoholic beverage guys.
