There are really no "to-go" boxes at any of the counter service restaurants. There are desserts that come with lids in many locations, but as far as entrees go, it is very slim. France's Boulangerie Patisserie in the WS at EPCOT does have some boxes for their stuff to go in. You can also find some prewrapped sandwiches in many of the resort food courts. I have not eaten at DTD, so I do not know about the restaurants there. Hopefully someone will chime in with their DTD experience.
At the Earl of Sandwich in DTD all of their sandwiches and sides are already packed to go. They are wrapped or in little plastic containers. Even the soup comes in a bowl with a lid. They hand it all to you in a bag.
In the past I have taken out a Chicken Ceasar Salad from the Contempo Cafe @ Disney's Contemporary Resort. When it was ready I asked a Cast Member for a lid since I said it was take out and go it without a problem, along with a bag to put in.
We have taken out from Tangierine Cafe (Morocco). They had styrofoam to-go clamshell boxes. We have also taken food to go from Sunshine Seasons, although they were initially reluctant because they didn't have proper containers. What they ended up doing was putting one plate over the other plate and wrapping the dinner in Saran.
One thing we discovered is that the places do not have large bags to carry everything. Once, we were able to get a large shopping bag from the Soarin' gift area. After that, we started brining our own large bags when we are planning on taking out.
You can always get counter service to go. The question is whether or not you can get it packaged for carrying out. A lot of the park CS do not provide to-go packaging, so you have to carry the food on paper plates. You are more likely to be able to get things packaged at DTD or resorts.