Most construction and cleanout debris is fine. The limits come from Mississippi's disposal site rules and from the facility that takes your load.
The Mississippi Department of Environmental Quality sets clear rules for the disposal sites that take only construction debris. According to its debris site guide, those sites accept wood, metal, brick, concrete, roofing, furniture, and cardboard. They refuse household garbage, food waste, liquids, paint cans, refrigerators, engines, whole tires, batteries, and asbestos.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos also can't go in a standard dumpster.
For household chemicals, electronics, and tires, watch for the City's yearly Right Way to Throw Away collection day. If you're unsure about an item, ask your account manager before loading it.