Most construction and household material is fine. The limits come from Nebraska law and from the transfer station's own rules, and one of them is seasonal.
Nebraska bars land disposal of lead-acid batteries and waste oil outright, and bars waste tires in any form unless the tire cannot be recycled. Yard waste is the seasonal one: a landfill may take it without condition only from December 1 through March 31, and the rest of the year it has to be separated at the source for composting or soil conditioning. For an April to November cleanout that mixes brush with household junk, that means two containers or a separate trip for the green material.
The Sarpy County Transfer Station adds its own prohibited list: asbestos, corrosive material, flammable and explosive waste including lithium-ion batteries, fluorescent tubes and anything with mercury, pesticides and herbicides, solvents, thinners, paints and inks, petroleum-soaked soil, treated railroad ties and power poles, waste left in drums or buckets, and bulk liquids. Waste oil and lead-acid batteries are not accepted at all. Tires and major appliances are taken only at separate drop-off points for an extra charge, and heavy equipment tires are refused.
Concrete is accepted, with a catch worth knowing before you load: it has to be broken down to pieces no larger than two feet.
Anything you are not sure about, ask your account manager before it goes in. One refused item can hold up a whole load at the gate.