Most construction and cleanout debris can go in. The limits come from Texas landfill rules and from City facility rules.
The Texas Commission on Environmental Quality prohibits used oil, used oil filters, lead-acid batteries, and regulated hazardous waste at every type of municipal landfill, according to its guide to waste accepted by landfills. Keep car batteries and oil out of your container.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster either.
Heavy masonry needs extra thought. The City's J.C. Elliott Transfer Station takes lumber, sheetrock, roofing, and dirt from homeowners, but not rock, concrete, or brick. Tires carry their own per-tire fee there. Mention masonry and tires when you book.
Your account manager can confirm whether an item is allowed before it goes in.