Most debris from a remodel or cleanout can go in your container. New Jersey's disposal rules and the facility that receives your load set the limits.
New Jersey bans throwing away TVs, computers, laptops, and computer monitors. The state's electronic waste rules have required these items to go to collection centers since January 1, 2011. Pull them out before you load.
Hazardous waste, liquids, paint, solvents, fuel, motor oil, antifreeze, gas cylinders, and asbestos can't go in a standard dumpster either. Bergen County runs household hazardous waste collection events for items like these. Those events don't take tires, electronics, pressure-treated lumber, or oxygen and acetylene tanks, but the county holds separate electronics and tire collection events.
Plan to handle tires and treated lumber on their own, and mention them when you book.
Ask your account manager about anything you're unsure of before it goes in the container.