Most remodeling and cleanout debris is fine to load. The limits come from Michigan's landfill rules and from what the receiving landfill will take.
The Michigan Department of Environment, Great Lakes, and Energy keeps a list of materials that can't go into a regular landfill. It includes scrap tires, used oil, lead acid batteries like the kind in a car, bulk liquids, returnable beverage containers, hazardous waste, and medical waste. Asbestos and PCB waste can only be landfilled under special handling rules.
Yard waste is on the list too. As the City puts it on its garbage page, Michigan has banned yard waste from landfills since 1995. Keep grass, leaves, and brush out of your container and use the City's seasonal yard waste pickup instead.
Paint, solvents, fuel, and other hazardous liquids stay out of a standard dumpster as well. Tires are refused at the curb and at the landfill, so set them aside.
When you aren't sure about an item, ask your account manager before it goes in.