Yes they do have a "dress code" but I use that term very lightly, since they rarely enforce it, and it's not even really what I would really consider a dress code.
Taken from the Disney website;
Dress Code Guidelines:
Men: Khakis, slacks, jeans, dress shorts, collared shirts. Sport coats are optional.
Ladies: Capris, skirts, dresses, jeans, dress shorts.
NOT permitted in dining room: Tank tops, swimwear, hats for gentleman, cut offs, or torn clothing. While t-shirts are now allowed, the policy remains that t-shirts with offensive language and/or graphics are not acceptable.
I eat at several signature restaurants every trip, and I almost always wear capris, nice sandals, and a nicer top then I would in the park, but nothing too fancy.