Framing, plaster, lath, drywall, roofing, flooring, cabinets, doors, windows, siding, fencing, furniture, and general cleanout debris are all fine in a standard container.
Hazardous material is not. Paint, solvents, fuel, pesticides, pool chemicals, motor oil, and asbestos all have to stay out, along with liquids of any kind. Alliance's own rules treat spillage or drainage from a load as an offense, which is why containers travel covered and why wet or liquid waste is refused.
Tires, electronics, appliances, and household hazardous waste need their own route. The county recycling district runs a drop-off site in Canton that takes those items from residents of Stark, Tuscarawas, and Wayne counties, but it is appointment only, it caps tires at ten per person per year, and it does not accept latex paint. For a construction load, tell us about these items up front and we will price them separately.
Asbestos deserves its own mention in a town with this much pre-war housing. Pipe wrap, old floor tile, siding shingles, and some plaster systems can contain it, and it has to be surveyed and removed by a licensed abatement contractor rather than going in a container.
Anything you are unsure about should go past your account manager before it is loaded.