Everyday construction, demolition and household debris is all acceptable. The exclusions here are mostly set by Ohio law and by what the local facilities will accept.
Liquids are the firm one. The city states that under state law liquids cannot be accepted at transfer stations, and that the Strongsville station accepts liquids of no kind at all. Paint, gasoline, oil, thinners and solvents all stay out of the container.
Whole tires may not go to landfill anywhere in Ohio. Locally they are accepted only off the rim and only at the transfer station, and only from residents, so on a job they need their own route.
Asbestos is prohibited outright. On a Strongsville house built before the mid-seventies, pipe lagging, floor tile and some siding are the usual places it appears, and finding it in the first week is far cheaper than finding it at the gate.
Refrigerated appliances need their refrigerant removed before disposal. For curbside collection the city requires it to be taken out by a certified professional and the appliance tagged, with freezer and refrigerator doors removed.
Also excluded locally: dead animals, poisons, explosives, hazardous and corrosive chemicals, grass clippings and leaves. Soil, rocks, bricks and concrete blocks are excluded from the transfer station specifically, which is another reason heavy fill belongs in a container with a destination arranged in advance.
Carpeting has a local shape worth knowing if any of it is going curbside instead of in the container: cut into three-foot sections, rolled and tied, and bundles kept under 35 pounds.
If anything is borderline, one call to your account manager settles it first.