Lumber, drywall, shingles, siding, flooring, cabinets, fencing, furniture, and general junk all load without trouble. The real limits are hazardous material and the County's own rules at the gate.
Hazardous waste never goes in a container. Paint, solvents, fuel, motor oil, antifreeze, pesticides, and asbestos are all out, and so are liquids and sealed drums. The County keeps a separate recycling area at the landfill for tires, paint, antifreeze, oil, metal, and brush, and it asks customers to sort those out of the load rather than bury them.
Alabama splits its waste duties between agencies, with the state environmental agency permitting disposal sites and the public health department covering household collection, so there is no statewide banned-materials table to work from. What governs your load is the receiving site, and the County bills tires by the ton whatever their size.
The City's own collection rules are stricter than a container. City crews will not pick up tires, rocks, dirt, concrete, block, brick, roofing material, construction waste, sheetrock, paint cans, pesticides, or batteries at the curb at all. One useful exception is clean wood: the City will carry away stacked tree trunks, limbs, and other wood products for its own wood waste operation, provided the pile holds nothing but wood and is free of any other debris. Anything you are unsure about should go past your account manager first.