Nearly all League City construction and household debris is accepted. The limits come from the receiving site's own rules rather than from a single state list, and Texas sorts disposal sites by what they are allowed to take.
That sorting is worth understanding. Texas has a class of site that may only accept brush, construction and demolition debris, and similar non-household material, and may not take household waste at all. A clean construction load can therefore go to a site a mixed household load cannot, which is one reason keeping a renovation container free of kitchen waste can matter.
Hazardous material, liquids, paint, solvents, fuel, and asbestos are excluded from every standard container.
The nearest public drop-off site in the area publishes a useful refusal list of its own: it accepts building materials and roofing materials, but refuses concrete, tires, combustible materials, aerosol cans, freon, gasoline, kerosene, and paint. Concrete being on the refusal list is worth noting if you are taking out a slab, because that load needs a destination chosen for it.
Tires and appliances go on the booking sheet. Ask your account manager about any doubtful item before it goes in.