A good idea is that at the parks, proper park attire is appropriate at at TS restaurants. I would say that any TS restaurants should mean being neat and clean. If you are going to a TS at a resort like Artist Point or California Grille, attire is business/resort casual....no shorts and me should have a collared shirt. No shorts, T-shirts. It is always better to lean toward too "dressed up" than not. A good place for information is www.allearsnet.com.
Uncleremus has generously posted the official dresscode for the Disney Signature restaurants. Basically, according to Disney you can wear just about anything accept swimwear and tanktops and hats for men so there are really no worries.
In my opinion, I would feel uncomfortable in any of the signature restaruants in a t-shirt, jeans or flip flops. These restaurants are very nice and I feel more comfortable in nice pants and a collared shirt. It also makes the dinner a little more special. Rarely, if it is hot my kids will wear a nice pair of shorts and a collared shirt but that is as casual as we will go.
If you search you will find some lively discussion on this topic.
I have to agree with what is posted. I like to dress up a bit when I go to the resorts. During the summer my husband will wear shorts with a nice silk shirt, that is about as casual as we will get.