Yes, some of them do. Mainly the signature restaurants like Narcoossee's, Yachtsman Steakhouse, V & A's, etc. When you make your reservations they will tell you about the attire.
Signature restaurants only. It boils down to don't wear swimwear, ratty shorts, torn clothing or offensive tee shirts. If you are a guy, don't wear a tank top or a hat.
And the dress code has nothing to do with the DDP at all.
DDP is the Disney Dining Plan and it's still in existence. The DDE or Disney Dining Experience (discount card) is now Tables in Wonderland or TiW. Clear as mud?
Nala is right, how you pay has nothing to do with the dress code of the restaurant.
Dress codes are suggested at all signature locations and do boil down to basically the requirements Nala gives. Victoria and Albert's (who only accept the Premium or Platinum Dining Plans) has a strict dress code. Jackets for men and dressy clothes for women.
Unfortunately, too few people make an effort to dress properly for dinner; and too few restaurant managers are willing to turn anyone underdressed away, or ask gentlemen to remove their baseball caps while they eat (even though a gentleman should know better in the first place).