Normal construction and household debris is welcome: framing, drywall, plaster, lath, roofing, siding, flooring, cabinets, doors, carpet, furniture and cleanout junk.
Hazardous material is not. Paint, solvents, thinners, fuels, motor oil, pesticides and pool chemicals all stay out, together with asbestos, free liquids and pressurized cylinders. Houses near Ely Square are old enough that asbestos turns up in floor tile and pipe wrap, so test before demolition.
Tires come with a state rule attached and it is stricter than most people expect. The Lorain County Solid Waste Management District states on its own district page that in Ohio it is illegal to transport more than ten scrap tires at a time anywhere in the state without first being registered with, or holding a registration certificate from, the Ohio Environmental Protection Agency. Keep tires out of the container and handle them separately.
Appliances holding refrigerant must have the refrigerant removed by a certified program before disposal, which again is easier arranged before the container arrives.
Dumping inside the city is barred outright. Elyria's code makes it unlawful for any person to dispose of, bury, burn or dump any garbage or household trash within the city limits, wherever it came from. When an item is borderline, ask your account manager.