Lumber, drywall, plaster and lath, slate and asphalt roofing, flooring, cabinets, doors, fixtures, furniture, and general cleanout debris are all fine in a roll-off container.
The firm exclusions are the usual hazardous ones. Keep out paint, solvents, fuel, motor oil, antifreeze, pool chemicals, pesticides, batteries of every type, fluorescent lamps, and electronics. Liquids of any kind are out, and so are propane tanks and other pressurized cylinders. Tires need separate handling.
Asbestos deserves its own paragraph in a city built mostly before 1940. It turns up in pipe insulation, old floor tile and its adhesive, siding, and furnace rooms across the Shaker Heights housing stock, and it has to be surveyed and removed by a licensed abatement contractor rather than loaded into a container.
Ohio does not publish a single statewide banned-materials table, so the practical limits here come from what the receiving facility will accept and from the hazardous waste rules rather than from a city list. Tell your account manager what the job involves and anything that needs separate routing gets arranged alongside the container.
One Shaker Heights specific: the City's residential collection rules cap household containers at 32 gallons and 40 pounds filled. That is about curbside cans rather than roll-off service, but it explains why renovation debris has to leave by container rather than by the weekly collection.