Most remodeling and cleanout debris is fine. The limits come from Ohio's landfill rules and from what the county landfill accepts.
The Franklin County Sanitary Landfill turns away hazardous waste, asbestos, liquids, motor oil, car batteries, gas cylinders that aren't empty, and uncleaned drums or buckets. It also refuses TVs, computers, and other electronics, plus tires, yard waste, and appliances that still hold Freon.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster either.
Latex paint is the one exception to the paint rule at the landfill, but only once it has fully dried. Leave the brush, grass clippings, and leaves out of your container, since yard waste isn't landfilled here.
If you're not sure about an item, check with your account manager before you load it.