Check the National Parks Website for regional parks and their individual rules. There are a number of parks where you can get Backcountry Camping Permits, but most will not allow fires except at certain designated camping areas.

Also, check the Appalachian Trail Conservancy Site for trails and backcountry shelter locations, which are open and free for hikers to use on a first-come first-serve basis. You can have fires at many of the shelters because they have established fire rings. There are also a number of sites where you can pitch a tent that have fire rings of there own.

I know here in Virginia they have "Dispersed Camping" where you can pick any spot to camp as long as you follow the "Leave No Trace" principles and it doesn't look as if the spot is overcamped. In a few almost rare places campfires are allowed if it hasn't been a very dry season.