Framing, drywall, shingles, siding, flooring, cabinets, doors, trim, carpet, fencing, furniture and ordinary cleanout junk are all fine to load.
Ohio bars whole tires from landfill outright. They have to be recycled instead, and Greene County Public Health points people to the transfer stations in neighboring Montgomery County or to haulers that take them, so tires need their own arrangement and their own charge.
The health district frames the rest as a duty to use an approved facility: trash, garbage, tires, infectious waste and other debris, construction and demolition debris included, must be properly disposed of at an approved facility. In practice that keeps the usual material out of every container here, so no hazardous waste, no liquids, no paint, solvents or fuel, no used oil or antifreeze, no batteries, no medical waste and no asbestos.
Fill is its own question in Greene County. Only clean, hard fill may be used to raise a low spot on your land, ground inside the hundred-year flood plain may not be filled or disturbed at all, and the health district publishes a fact sheet on what counts. If you are moving soil, concrete or masonry rather than throwing it away, mention it to your account manager, because the cheapest route may not be a container at all.