Lumber, drywall, plaster, lath, slate, tile, flooring, cabinets, fixtures, siding, furniture and general household contents all load without trouble. The limits come from Ohio law, from Cuyahoga County rules and from the receiving facility.
Ohio bars whole tires from landfill disposal by name; they must be recycled instead. Hazardous waste, liquids, paint, solvents, fuel, used oil, asbestos, propane cylinders and medical sharps are refused in every container placed. Appliances containing refrigerant need separate handling, and electronics belong at an electronics outlet rather than in a mixed load.
Cuyahoga County adds a rule that matters on a demolition job. Its demolition standards say construction and demolition debris must leave the property and go to one of four places. Those are a state licensed construction debris landfill, a state licensed solid waste landfill or transfer facility, a certified recycling plant, or, for clean hard fill, the state fill rules. The same standards bar debris from being stored on site or allowed to spread onto nearby land or into the public right of way.
A practical consequence for Cleveland Heights customers: the City's curbside service will not take your project debris at all. Its bulk collection expressly excludes large move-outs, clean-outs and any building or construction debris. If a project generates it, a container is the route.
Anything you are unsure about, raise it with your account manager before it is loaded.