Most construction and cleanout debris is fine. The limits come from the city's own prohibitions and from what the county facility charges or refuses.
Hazardous material is out of any container for regular collection. The city names hazardous material, radioactive waste, biomedical waste, flammable material, explosive material, and anything else dangerous or highly corrosive, and puts the cost of proper disposal on whoever generated it. Paint, solvents, fuel, used oil, antifreeze, and pool chemicals all fall in that group.
Tires are specifically called out. City code prohibits placing automotive, motor vehicle, or trailer tires for collection, and the county charges for them by the piece or by the ton. Propane tanks, cylinders, fire extinguishers, and oxygen tanks are each priced per item at the county facility, so they belong in their own arrangement.
Two more city rules affect how a load travels. Solid waste that could leak or spill may not be carried on a public road unless it is securely tied or covered, and nothing may be allowed to blow or drop from a container. In practice that means a covered load, every time.
Household hazardous material and electronic waste are accepted at the county facility at no charge, which is a far better route than a container. If an item has you guessing, ask your account manager before it goes over the rail.