Most remodel debris, yard waste, and household junk can go in your dumpster. The limits come from Texas landfill rules and from the landfill that takes your load.
The Texas Commission on Environmental Quality keeps several items out of every landfill in the state. That list includes car batteries, motor oil you changed yourself, used oil filters, whole tires, and anything with refrigerant, like a fridge, freezer, or window air conditioner. Liquid waste and regulated hazardous waste are banned too.
Paint, solvents, fuel, and asbestos don't belong in a standard dumpster either. The Melissa landfill also turns away loader, tractor, and grader tires.
The Town's curbside bulk pickup has its own limits. It won't take bricks, concrete, tires, or car parts, and any fridge or AC unit needs a technician's tag showing the refrigerant was removed. That's one reason a dumpster is the easier route for remodel debris.
If something on your list gives you pause, ask your account manager before it goes in the container.