Most construction and cleanout debris can go in your container. Texas rules and the local disposal site decide what can't.
Texas doesn't let whole tires go into a landfill as they are. They have to be split, quartered, or shredded first, and landfills usually charge extra for them. Lead-acid batteries, like car batteries, can't be thrown away in Texas at all. Stores that sell them have to take used ones back for recycling.
The City's Transfer Site sets its own limits. It won't take household garbage, recyclables, electronics, or hazardous waste such as paint, pesticides, and chemicals. It takes up to six tires per visit, and each must be no more than 25 inches across.
Beyond that, leave liquids, solvents, fuel, pressurized tanks, and asbestos out of a standard dumpster. Mention tires and appliances when you book.
If you're not sure about something, ask your account manager before you toss it.