Lumber, drywall, plaster, roofing, flooring, carpet, cabinets, fixtures, furniture and general cleanout debris are all ordinary roll-off material.
The Wadsworth specific is not a legal ban but a destination limit, and it is the most useful thing here. The City's own transfer station publishes its refusal list, and construction materials and shingles are both on it, alongside brush, yard waste, tires and liquid waste. So the material from almost any Wadsworth building job cannot go to the city station at all, whoever takes it there. That is why the container and its destination are arranged together.
The ordinary exclusions apply as well. Liquids, paint and solvents, fuel, motor oil, antifreeze, pressurized cylinders and asbestos stay out of any Wadsworth container. Asbestos is a real consideration in Wadsworth's older housing, where it appears in pipe wrap, floor tile and its adhesive, and textured ceiling coatings. It needs a licensed abatement contractor and its own route.
Tires are the item most often forgotten on an Ohio cleanout, and between the state ten-tire threshold and the city station's refusal they need planning rather than improvisation.
Electronics need a separate stream too, which matters on an office or basement clearance where furniture and hardware come out together.
Tell your Wadsworth account manager what the job involves and roughly when the building went up, and the separate routes get arranged alongside the container.