Most household and construction debris is accepted. The limits come from Texas rules on what a landfill may take.
A Texas landfill can't accept lead-acid batteries, used motor oil, used oil filters, whole scrap tires, bulk liquids, or appliances holding refrigerant that wasn't removed properly. Tires can go to a landfill only after being split or shredded, so keep them out of the container.
The familiar exclusions apply too. Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster.
Two local notes help. City bulky waste pickup won't take construction or demolition debris, electronics, or hazardous material, which is why a remodel needs a container. And heavy clay soil belongs in its own small load rather than mixed with light debris.
If an item has you second guessing, ask your account manager before it goes in.