Nearly all household and construction debris is accepted. The limits here come from Ohio law and from the county facility's own rules.
Ohio bars whole tires from landfill by name, and they have to be recycled instead. The state also requires a hauling license for anyone moving more than ten scrap tires, which is why tires get handled separately rather than tossed in with a tear-off.
Hazardous waste is the other hard line. Paint, solvents, fuel, used motor oil, antifreeze, pesticides, pool chemicals and automotive cleaners all need their own route, and the county runs collection events for exactly that. Asbestos, liquids and anything flammable stay out of a standard container anywhere.
Appliances, electronics and scrap metal are all recyclable through county programs rather than being buried, so pulling them out before the container arrives usually saves weight and money. Yard waste has its own route too, and the City collects brush separately from trash.
If an item sits somewhere in between, ask your account manager before it goes in.