Framing, drywall, roofing, flooring, cabinets, trim, siding, fencing, concrete and ordinary household contents all load fine. Tennessee publishes no statewide banned-materials table, so the limits here come from the receiving landfill and from the City's own list.
The City sets out plainly what its crews will not pick up, and the list doubles as a guide to what needs its own handling: tires, gas cans, propane tanks, concrete, hazardous waste such as chemicals, paint cans, medical supplies, major demolitions and lot clearing. The same page is blunt about contractor debris: anything left by a contractor, including a remodeler or a tree trimmer, must be removed by that contractor.
Keep out of any Cleveland container: paint, solvents, fuel, used oil, pool chemicals, pesticides, batteries, propane cylinders, medical sharps, asbestos and liquids of any kind. Appliances holding refrigerant and whole tires need separate handling and should be declared when you order rather than discovered at the gate.
Brush is the local trap. Cleveland forbids mixing metal, lumber, brick, rock or dirt in with brush trimmings, and forbids mixing leaves and grass clippings with other brush. If your job produces both vegetation and construction debris, plan on two containers.
If you are unsure about an item, ask your account manager before it goes in.