Most household and construction debris can go in. The limits come from Texas rules and from the landfill that takes your load.
Tires are the main state rule to know. The Texas Commission on Environmental Quality says scrap tires can go to a permitted landfill only once they've been split, quartered, or shredded. Keep whole tires out of the container.
Hazardous waste, liquids, paint, solvents, fuel, motor oil, propane tanks, and asbestos can't go in a standard dumpster either.
The City's own drop-off site is stricter. The Marshall Convenience Station won't take roofing, dirt, rock, brick, concrete, large appliances, water heaters, air conditioners, batteries, tires, oil, or paint. That's most of what a remodel produces, which is why a container is usually the better fit.
If you have something you're unsure about, ask your account manager before it goes in.