Most household and remodeling debris can go in your container. The limits come from the landfill that takes your load and from basic safety rules.
Brown Station Road Sanitary Landfill won't accept animal carcasses, asbestos, ash, automobiles or car parts, compressed gas cylinders, hazardous waste, liquids, or radioactive material. It also refuses soil, rocks, stumps, logs, shingles, and roofing material.
Construction debris has its own limit there. The County allows each household one non-commercial pickup or trailer load of construction and demolition material per year. Larger remodel loads need a permitted construction debris facility instead.
Paint, solvents, fuel, and other chemicals don't belong in a standard dumpster either. Tires and appliances should be kept separate and mentioned when you book, since the landfill handles them apart from trash.
If you're unsure about any item, ask your account manager before it goes in.