Framing, drywall, cabinets, flooring, shingles, furniture, and general cleanout junk are all fine. The limits come from Tennessee's disposal rules and from what the county station will take off the truck.
Tennessee keeps whole tires out of its landfills. Under the state's waste tire program, a whole, unshredded tire has to go to a permitted processor or a county tire collection site instead. Every Tennessee county runs at least one of those sites.
The Rutherford County Transfer Station adds its own list of refusals. Drums, tires, concrete slabs, asbestos, batteries, hazardous waste, sludge and liquids, and oversized material are all turned away there. Staff inspect loads by camera and ask the driver what's in the trailer, and a load with the wrong material in it goes back on the truck at the hauler's cost.
Hazardous waste, liquids, paint, solvents, and fuel can't go in a standard container anywhere. Neither can asbestos. Mattresses, tires, and large appliances need to be handled on their own.
One questionable item can hold up a whole load, so run anything unusual past your account manager before it goes in.