Lumber, drywall, roofing, siding, flooring, cabinets, fencing, furniture, and general cleanout material are all fine. The limits come from Texas rules on certain materials and from what Tyler keeps out of its own containers.
Texas regulates scrap tires on their own track. Transporting, storing, processing, and recycling them all fall under the Texas Commission on Environmental Quality, which is why tires never ride along in a dumpster.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos stay out of a standard container, and so do appliances, pesticides, and other chemicals.
Tyler's own rules add useful detail. The landfill is set up for only small amounts of household hazardous material, and the city asks that paint be fully dried before it is thrown out. Automotive, lawnmower, and tractor batteries, motor oil, and antifreeze go to the city's recycle center downtown instead. Fluorescent bulbs are handled separately too.
Still unsure about something in the pile? Ask your account manager before it goes in the container.