Framing, drywall, shingles, siding, flooring, cabinets, doors, carpet, fencing, furniture and ordinary cleanout junk can all go in the container.
Tennessee controls disposal through what a permitted landfill may accept rather than through a single statewide banned list, and anything classed as special waste needs its own approval from the state before a landfill can take it. In practice that means the usual exclusions apply to every container: no hazardous waste, no liquids, no paint, solvents or fuel, no asbestos and no medical waste.
The City is specific about the household items it will not take in a cart, and the same items cannot ride in a container: gasoline, liquid paint, pool chlorine, batteries, motor oil, antifreeze, pesticides and transmission fluid. Those belong at the Memphis and Shelby County household hazardous waste site instead.
Tires and large appliances need separate handling and their own charge, so mention them when you book. Anything you are unsure about is worth a quick word with your account manager before it goes in.