Everyday household and construction debris is fine. The exclusions come from Pierce County disposal rules and from what the receiving station will accept at the gate.
Pierce County is blunt about hazardous material. Its household hazardous waste page says these products should never be poured down the drain, onto the ground, into storm sewers or thrown in the trash. That covers gasoline, antifreeze, automotive fluids, oil-based paint, paint thinner, pesticides, pool chemicals, aerosol cans and propane tanks.
The County will not take asbestos, ammunition, explosives, compressed gas cylinders, biological waste or radioactive material at its hazardous waste facilities at all. Those need a specialist, and none of them belong in a roll-off container.
Electronics, motor oil and vehicle batteries are handled separately at the transfer stations rather than buried, so pull them out before pickup. Businesses have their own route for hazardous material through the Tacoma-Pierce County Health Department.
Tires, mattresses and large appliances usually carry a charge of their own. Tell your account manager about them when you book and the quote will reflect it.