Standard construction and cleanout material is fine: lumber, drywall, roofing, flooring, cabinets, doors, windows, fixtures, brick, block and household goods.
Nothing hazardous goes into an Olive Branch container, and nor do liquids, paint, solvents, fuel or asbestos.
The City's own code gives a useful list of what must never go into a collection cart: gasoline, solvents, hot ashes, dirt, bricks, concrete, leaves, limbs, grass cuttings and garden trimmings. A roll-off will take brick, concrete and dirt, which is exactly why those materials end up in one, but the list shows which items need declaring when you book because they are heavy or routed separately.
The City also makes it unlawful to dispose of waste material anywhere other than a disposal site the City permits, so the destination is not a free choice. That is one more reason the container route is simpler than a trailer and a weekend.
Tires, appliances and electronics are usually priced per item rather than by weight, so set them aside and mention them at booking.
When a single item is the thing you are unsure about, your account manager can check it against the receiving landfill's own acceptance rules before the container arrives.