Nearly all Las Cruces construction and household debris loads normally. New Mexico publishes no single statewide table of banned materials, so the operative limits here are the regional authority's own acceptance rules and the ordinary hazardous exclusions.
Hazardous waste, liquids, paint, solvents, fuel, used oil, and asbestos stay out of every standard container.
The authority runs a separate household hazardous waste route and a separate recycling route, so paint, chemicals, batteries, and electronics belong in those rather than in your container. Tires and large appliances should be kept apart and named at booking, because they are usually priced per item rather than by weight.
The City adds its own duty on the loading side: a construction site has to be kept free of uncontained accumulations of wood, brick, sheetrock, shingle, cement, concrete, cinderblock, and asphaltic concrete, along with the wrappers and containers those products arrive in. In practice that means the container is where all of it should go as it comes off the building.
An item you are unsure about deserves a quick word with your account manager before it is loaded.