You can load most construction and household debris. The limits come from Texas landfill rules and from the landfill that takes your load.
Texas keeps several things out of municipal landfills. Car batteries, used motor oil, and oil filters aren't allowed. Whole tires must be cut or shredded before a landfill can take them. Refrigerators and freezers need their coolant handled properly first. Liquids, PCB waste, and regulated hazardous waste are also banned.
Beyond those rules, keep hazardous waste, liquids, paint, solvents, fuel, and asbestos out of a standard dumpster.
If you live in Fort Worth, the City's drop-off stations take household chemicals, paint, car fluids, and batteries for free. Bring proof of residency. Those sites won't take commercial loads, concrete, dirt, tile, or roofing, which is where a roll-off helps.
When you're unsure about an item, ask your account manager before it goes in the container.