Most construction and cleanout debris is fine. Texas does not publish a statewide list of banned materials the way some states do, so the limits come from the landfill that receives your load and from the city's own definitions.
Hazardous waste, liquids, wet paint, solvents, fuel, pesticides, and asbestos stay out of any standard container. The city treats anything radioactive, corrosive, highly flammable, explosive, or medical as waste its crews will not touch, and the same logic applies to a rented container.
The receiving landfill adds its own refusals. Lead-acid batteries, used oil filters, liquids, pesticides, refrigerants, and whole tires are all turned away, and tires are only accepted when they have been quartered. Household hazardous waste such as leftover paint and old medication is not allowed either.
Soil is its own category. The city's rules treat dirt, rock, and sand used as fill as unacceptable waste, and the landfill takes clean soil only on weekdays and only by appointment. If your job produces a pile of dirt, say so when you book so it can be handled separately.
One Midlothian quirk worth knowing: clean concrete, dirt, rock, and old asphalt are not counted as construction debris under the city's definition. They are a separate material, and they usually cost less when they travel in their own load. Check anything unusual with your account manager first.