Ordinary house and job-site debris is fine. The limits come from Texas rules and from what the receiving facility will take.
Texas keeps three things out of every landfill in the state, whatever its class: used motor oil and used oil filters, lead-acid batteries, and regulated hazardous waste. The state environmental agency lists all three as prohibited, which means they need a separate route no matter who hauls them.
Electronics are a local sticking point. Leander's curbside service won't take televisions or computer monitors at all, and the nearest city recycling center turns away televisions too. Pull them out of a cleanout before the container arrives.
Appliances split two ways. A water heater, stove, washer, or dishwasher can be recycled nearby, but anything with Freon in it, such as a fridge or a window air conditioner, has to be handled separately.
The usual exclusions round it out: paint, solvents, fuel, antifreeze, pool chemicals, propane tanks, and anything containing asbestos. Liquids of any kind are out as well. Not sure about one item? Ask your account manager before it goes in.