Most remodeling and cleanout debris can go in your container. The limits come from Texas landfill rules, which apply at every municipal landfill in the state.
The Texas Commission on Environmental Quality bans these items from municipal landfills: lead-acid batteries, used motor oil from home oil changes, used oil filters, and whole tires. Fridges, freezers, and air conditioners that hold CFC refrigerant are on the list too. Liquid waste, regulated hazardous waste, PCBs, and radioactive material are also banned.
Tires need extra care. Texas lets a landfill take tires that have been split, quartered, or shredded, but not whole ones. Keep whole tires out of your dumpster and mention them when you book.
Paint, solvents, fuel, and asbestos stay out of a standard dumpster as well. For household chemicals, the City's Who to Call list gives 972-440-2040 for hazardous waste pickup.
If you aren't sure about an item, ask your account manager before it goes in.