Most household and construction debris loads without any trouble. The limits come from Texas disposal rules and from what the facility receiving your load will take.
Texas keeps a short list of items out of every municipal solid waste facility in the state. Lead-acid batteries, do-it-yourself motor oil, used oil filters, and whole tires are all on it. So are refrigerators, freezers, air conditioners, and anything else holding chlorinated fluorocarbon coolant. Liquid waste is out as well.
Hazardous waste, paint, solvents, fuel, and asbestos can't go in a standard container either.
Electronics deserve their own look. The Texas Commission on Environmental Quality notes that discarded TVs and computer monitors may count as hazardous waste, and it asks that used electronics be recycled instead. Pull them out before loading day.
Local facilities add a few refusals of their own. The transfer stations take tires only through a resident drop-off program, and the landfill won't accept loader, farm tractor, grader, or scraper tires. Tell your account manager about tires or large appliances and they'll arrange a separate route for them.