Everyday Licking County renovation, roofing and cleanout debris is fine. The limits come from the receiving facility and from the usual hazardous categories.
Liquids stay out, and so do paint, solvents, thinners, fuel, motor oil, antifreeze, pool chemicals and pesticides. Appliances holding refrigerant, fluorescent tubes, batteries, televisions and computer equipment all have separate routes.
Tires deserve their own mention in Ohio. The state licensing rules for moving scrap tires engage at ten tires, which is a lower threshold than most people expect and easy to pass on a garage or barn clearance. Keep them out of the container and handle them separately.
Asbestos needs testing and licensed handling, not a judgement made while loading. In Newark's pre-war housing it shows up in pipe and boiler lagging, floor tile adhesive, shingle siding and textured ceilings, so order the test before demolition starts.
Ohio does not publish a readable statewide banned-materials table, so we will not present one. What we do instead is confirm what the receiving facility for your specific load will and will not take, which is the answer that actually governs your job. Raise anything unusual with your account manager before it goes in.