Lumber, drywall, shingles, flooring, cabinets, doors, furniture, and plain household junk are all welcome. The limits come from Michigan's landfill rules and from the receiving site's own gate policy.
Michigan publishes a list of materials its landfills cannot take. The state's landfill prohibited materials list covers hazardous waste, bulk liquids, used oil, lead acid batteries, medical waste, asbestos, scrap tires, and yard clippings, among others. A few of those are worth knowing in detail.
Yard clippings are the one that surprises people. Grass, leaves, sticks, and landscape trimmings have to be composted or taken to a composting site, and only clippings that are diseased, infested, or part of an invasive species program may be landfilled. Keep them out of the container and out of your mixed load.
Tires are allowed at a landfill only after they're cut in half or shredded, so they come out before loading. Used oil has to be recycled in Michigan, not thrown away. Bulk liquids from a business have to be solidified before disposal, which in practice means no liquids at all in a rental container. Asbestos is handled under a federal air standard and only by arrangement with the landfill itself.
Paint, solvents, fuel, and pressurized cylinders stay out as well. Run anything unusual past your account manager before it goes in, since one bad item can turn away a full load at the gate.