Most construction and household debris is fine to load. Texas rules and the landfill that takes your load set the limits.
Texas treats scrap tires differently from other trash. The Texas Commission on Environmental Quality says scrap tires can go to a permitted landfill only after they've been split, quartered, or shredded. Keep whole tires out of your container.
Household hazardous waste has looser state rules than many people expect. TCEQ's household hazardous waste page says items like oil-based paint, pesticides, pool chemicals, and fuels from a home can often go in regular trash. A roll-off is a different case, though. Liquids, fuel, propane tanks, paint, solvents, asbestos, and other hazardous materials still aren't allowed in a standard dumpster, since they can get a load turned away.
Be careful with mixed loads. Household trash can't go to a construction-debris landfill at all, so food waste or bagged garbage mixed into a remodeling load limits where it can be dumped.
If you're unsure about an item, your account manager can tell you before it goes in.