Ordinary renovation and cleanout debris is fine. The limits come from the City's rules and from how Union County classifies waste.
Hazardous waste, medical waste, paint, solvents, fuel, used oil, and asbestos never go in a standard container. New Jersey licenses the companies that move those separately, so they travel on their own route.
The City's dumping rules also make it an offense to tip loose material onto any lot, dock, or pier in Elizabeth without the owner's written permission filed with the Public Works director, and the code presumes the owner of the vehicle used knew about it. Penalties include a fine and possible jail, so nothing leaves a job site except in the container.
Two container rules bite on content as well. A refuse container must not leak or discharge liquids into the City's storm drains, which rules out wet demolition slurry and standing water. And a permitted container cannot be treated as a dump in its own right: it has to be hauled to a licensed facility.
Recyclables take a different path. Elizabeth runs a mandatory recycling program, and the county's disposal control expressly leaves recyclable material out, so source-separated metal, cardboard, and clean wood are worth pulling aside. Anything you are unsure about, ask your account manager before it goes in.