Most remodel and cleanout debris is fine. The real limits here come from the county site that receives the load.
The county center refuses concrete, brick, blacktop, dirt, rock, household garbage, liquids, and anything wet. Used oil goes to the county road department instead. That refusal list is why a masonry demolition load and a drywall load are not the same job here, and why we ask what's going in before quoting.
Hazardous waste never goes in a standard container, and the local rules say plainly that the public collection service will not take it under any circumstances. Wet paint, solvents, fuel, pesticides, and automotive fluids all belong in that group, along with asbestos.
Medical waste has its own packaging rule locally, so it can never go in a container with other debris.
Tires and appliances are accepted at the county center but each one carries its own charge, so pull them out and mention them when you book rather than burying them in the load.
If you're unsure about something, ask your account manager before it goes in.