Most debris from a remodel, a roof job, or a cleanout is fine to load. The limits come from New Jersey's disposal rules and from what Bergen County's transfer stations accept.
New Jersey has banned throwing away TVs, computers, laptops, and computer monitors as regular trash since January 1, 2011. Pull them out before you load. The Borough collects this kind of electronics from residents at the Department of Public Works building behind Borough Hall.
Hazardous waste, liquids, paint, solvents, fuel, motor oil, propane tanks, and asbestos can't go in a standard dumpster either. The Borough's DPW crews also won't take tires, chemicals, wet paint, fluorescent bulbs, or car and rechargeable batteries at the curb, so plan a separate drop-off for those. Bergen County runs household hazardous waste collection events for residents.
If an item on your list gives you pause, ask your account manager before it goes in.