Most household and construction debris is fine to load. The limits come from Texas landfill rules and from the regional landfill that takes your load.
The Texas Commission on Environmental Quality lists wastes that can't go to any municipal landfill. They include lead-acid batteries, used motor oil, oil filters, and whole used tires. Refrigerators, freezers, and air conditioners containing refrigerant are on the list, too. So are liquid waste, regulated hazardous waste, and anything with PCBs.
Paint, solvents, fuel, and asbestos can't go in a standard dumpster either.
The landfill does offer a route for some of these. It takes up to four tires a load for a per-tire fee, refuses tires over 44 inches tall, and charges less for fridges and air conditioners when the refrigerant has been removed and certified. Leave those items out of your container and mention them when you book.
If you're not sure about something, ask your account manager before you toss it.