Everyday household and building debris is fine. The limits come from the type of landfill your load is sent to and from the materials no site will take.
Texas sorts its landfills into classes, and the class decides the contents. The Texas Commission on Environmental Quality is plain about it: a Type I landfill may accept all types of municipal solid waste, while a Type IV landfill may take only brush, construction and demolition debris and rubble. A Type IV site may not accept household waste or anything that rots. That matters to you because dropping a few bags of kitchen trash into a clean demolition load can send the whole container to a different, costlier destination.
Hazardous waste stays out of every container, everywhere. That means paint, solvents, thinners, fuel, motor oil, pool chemicals, pesticides, propane tanks and any sealed drum of liquid. Asbestos is licensed removal work and travels on its own route.
Tires, batteries, appliances holding refrigerant and large electronics are handled separately as well, and they are usually charged per piece rather than by weight, so name them when you book.
Unsure about a particular item? Run it past your account manager before it goes into the container.