Almost everything a remodel or a cleanout produces is fine. The limits come from Washington state law and from what the county facility will take at the scale.
Washington bars vehicle batteries from the trash entirely. They have to go back to a battery seller, a licensed collector, or a lead smelter, and a disposal site is not allowed to knowingly bury them. Vehicle tires are treated the same way, and can only be dropped at a site the state has designated for them. Both rules apply to you and to the facility, so one battery can hold up a whole load.
The usual suspects are out as well: hazardous waste, liquids, paint, solvents, fuel, used motor oil, and asbestos. Thurston County households can take hazardous material to HazoHouse at the county facility instead, and businesses book that by appointment.
A few items are accepted but priced per piece. Televisions, computer monitors, and anything with a compressor, such as a refrigerator, freezer, or window air conditioner, each carry an $18 handling fee on top of the garbage rate, and only the main county facility takes them. Asbestos has to be prepared and labeled to the County's posted requirements, and removing it needs clearance from the Olympic Region Clean Air Agency first. Passenger tires are capped at five per vehicle per day.
Not sure about something odd in the garage? Describe it to your account manager before it goes in.