Most construction and household debris is fine. The limits come from Oklahoma's rules and from city code, which is unusually detailed about this.
City code defines a long list of exempt waste that cannot go in any collection container: paints, stains and varnishes, adhesives, transmission fluid, gasoline and diesel, pesticides and fungicides, pool chemicals, household cleaners, propane tanks, camping cylinders, fire extinguishers, tires, vehicles and vehicle parts, and both automotive and household batteries. Hazardous waste is out, and so are explosives, poisons and corrosive chemicals.
Asbestos has its own rule statewide. The Oklahoma Department of Environmental Quality only allows regulated asbestos into landfills it has specifically authorized, and it publishes that list by county. Scrap tires go through the state's used tire recycling program instead of a landfill.
Two practical rules round it out. Anything mixed with water or other liquid has to be drained before it goes in, and grease has to be solid.
The city's annual free landfill coupon is worth knowing about but will not cover a project. It is for residential customers only, no commercial accounts or contractors, and it specifically excludes complete structures including roofs, demolished structures, trees, tires, batteries, hazardous material, wet paint and appliances that still hold refrigerant.
If an item is borderline, ask your account manager before it goes in the container.