Most remodel, roofing, and cleanout debris is fine. The limits come from Ohio's rules for transfer stations and from the facility that takes your load.
Ohio keeps three items out of regular loads at transfer stations. Scrap tires have to go to a tire facility. Loads made up only of yard waste have to go to composting. Lead-acid batteries, like car batteries, have to be recycled. Ohio law also bars mixing a used lead-acid battery with other trash.
The City's own trash service follows similar lines. It won't take batteries, tires, liquids, flammable items, or anything hazardous, and it sends residents to the county's solid waste district for those. Appliances that hold Freon need it removed by a certified technician first.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster either. Keep them separate and ask about drop-off options.
If you have questions about an item, your account manager can tell you before you load it.