Most remodeling, roofing, and cleanout debris can go in. The limits come from Texas landfill rules and from what the receiving landfill accepts.
The Texas Commission on Environmental Quality bars several items from every city landfill in the state. These include car batteries, used motor oil and oil filters, and whole tires. Fridges, freezers, and air conditioners that still hold refrigerant are also banned. Tires can go to a landfill only after they've been split, quartered, or shredded, so keep whole tires out.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster either. The same goes for anything with PCBs.
For household chemicals, the City sends residents to the Fort Worth Environmental Collection Center on Bridge Street. That site takes paint, batteries, and cleaners, but it won't take building materials or tires.
Not sure about an item? Ask your account manager before it goes in the container.