Most remodeling and cleanout debris is fine. The limits come from California's rules on hazardous and electronic waste and from what the receiving facility accepts.
The City says no hazardous waste or electronic waste may go in a temporary bin. That covers used motor oil and filters, paint, turpentine, strong cleaners, household and car batteries, pesticides, and weed killers. Electronics like TVs, computers, stereos, and phones belong in recycling.
Liquids, solvents, fuel, and asbestos can't go in a standard dumpster either.
For hazardous and electronic waste, Los Angeles County holds free collection events. You can find the next one by calling 888-253-2652, the county's Clean LA line listed on the City's bin page.
If you have an item you're unsure about, ask your account manager before you load it.