Almost all remodeling and cleanout debris can be loaded. The limits come from Tennessee rules and from what the county facility will take.
Tennessee makes it illegal to put wet latex paint in a landfill, and oil-based paint cannot go in the trash at all. Rechargeable batteries, including lithium and nickel types, have to be recycled rather than tossed. Those rules apply no matter who hauls the load.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos are out of a standard dumpster everywhere, and the county names several more. The transfer station turns away drums, tires, concrete slabs, asbestos, batteries, hazardous waste, sludge and liquids, and anything oversized.
Electronics need a separate stop. The county recycling centers will not take televisions, computers, paint, or fluorescent bulbs, and send electronic waste to the Haley Road site in Murfreesboro instead. Pull those items out before loading, and ask your account manager about anything you are unsure of.