Most remodeling and cleanout debris can go in your container. Ohio's landfill rules set the limits for the rest.
Ohio landfills can't accept hazardous waste, liquids, PCB waste, untreated infectious waste, or asbestos without the right permits. Whole and shredded scrap tires are barred at the landfill too. So is yard waste collected on its own, though a few branches mixed into a cleanout aren't the issue.
Ohio law also keeps lead-acid batteries, like car batteries, out of landfills, and stores that sell them take old ones back.
Tires carry an extra rule. Ohio EPA's scrap tire guide says anyone hauling more than 10 scrap tires in a load generally needs a state registration. For a handful of tires, Richland County residents can use the county's scrap tire program, which charges $3.00 per tire up to 18 inches, with a limit of 10 tires per visit.
Paint, solvents, and fuel stay out of a standard dumpster as well. The county Authority says a can that's less than a quarter full can be left open to dry, then thrown away once the paint has hardened.
If you aren't sure about an item, ask your account manager before it goes in.