Most debris from building, remodeling, or cleaning out a home can go in the container. Texas rules and the receiving landfill set the limits.
The Texas Commission on Environmental Quality bans several items from every municipal landfill in the state. They include lead-acid batteries, used motor oil from home oil changes, used oil filters, and whole scrap tires. Bulk liquids, hazardous waste, and PCB waste are banned as well.
On top of that, paint, solvents, fuel, asbestos, and propane tanks don't belong in a standard dumpster. Pull batteries, oil, filters, and tires out of the pile before loading, since a load that contains them can be turned away.
If anything on your list seems borderline, ask your account manager before it goes in.