Most building and household debris is accepted. The limits come from the receiving landfill and from Ohio's own rules.
The Franklin County landfill does not take yard waste, liquids or anything hazardous. Ohio bans whole tires from landfill by name, so tires have to be recycled instead, and state law keeps lead acid batteries out of landfill disposal as well.
There is also a commingling rule worth knowing before you load a demolition job. Ohio treats clean hard fill as a narrow category, covering only concrete, asphalt concrete, brick, block, tile and stone. Once other waste is mixed in with clean hard fill, all of the material has to be handled and disposed of as solid waste or as construction and demolition debris. That is the clearest argument there is for a second container: one for clean masonry, one for everything else.
Hazardous waste, liquids, paint, solvents, fuel and asbestos never belong in a standard container, and asbestos needs licensed abatement before demolition starts. The City also keeps freon appliances, car and motorcycle batteries, used motor oil, oil-based paint, tires, yard waste, explosives and radioactive material out of its own collection, and points residents to the county authority's free hazardous waste drop-off on East 8th Avenue in Columbus.
If an item is borderline, ask your account manager before it goes in the container.