Most construction and household debris is fine. The limits come from standard landfill rules and from what the receiving facility accepts.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos don't belong in a standard dumpster. Neither do propane tanks, gasoline, antifreeze, or pesticides.
The Wasatch Front Waste and Recycling District, which runs curbside service in Herriman, gives a helpful list of items to keep out of recycling carts. Its hazardous waste page names batteries, electronics, paint, gasoline, antifreeze, pesticides, full aerosol cans, propane tanks, and tires. Treat those items with the same care in a rented dumpster, and use the district's household hazardous waste drop-off instead.
Mattresses and tires often carry separate fees at local facilities, so mention them when you book.
If you aren't sure about an item, ask your account manager before it goes in.