Ordinary household and construction debris is welcome. The limits come from Oregon law and from what the county landfill accepts.
Oregon makes it illegal to put several things into any solid waste disposal site in the state. That list covers vehicles, large home and industrial appliances, used oil, tires, lead-acid batteries, and computers, monitors, and televisions. The rule reaches the person hauling the load, not only the landfill, so those items need their own route before your container is picked up.
Hazardous waste, liquids, paint, solvents, and fuel never belong in a roll-off container either. Household chemicals go to the county's hazardous waste program instead.
Asbestos gets special handling at the county landfill. It's taken by appointment only, and it has to be prepared and labeled to federal requirements with test results in hand. If your remodel touches old flooring, siding, or pipe wrap, say so early.
Unsure about an item? Ask your account manager before it goes in the container.