Ordinary construction and household debris is accepted. The limits come from Texas rules, from the district's own acceptance policy, and from one Richardson definition worth knowing.
Hazardous waste, liquids, paint, solvents, fuel, used oil and asbestos are out of a standard container everywhere, and Richardson's code adds a general prohibited category for anything a state or federal agency has named as unacceptable at a sanitary landfill.
Tires are the clearest line locally. At the district's transfer stations they are taken only through the resident drop-off program, not in a commercial load, and at the landfill they carry a separate per tire charge with large equipment tires refused outright. Keep tires out of the container and tell your account manager about them when you book.
Richardson also defines a category of debris its own crews will not collect, covering brick, lumber, plaster, concrete, large tree trunks, stumps, metal scrap, construction and remodeling materials and demolition or fire debris. That material is exactly what a roll-off container is for, and it is the reason a container rather than a city bin is the right tool for a remodel here.
Anything unusual is worth a quick call to your account manager before you load it.