Most household and construction debris is fine to load. The limits come from California's rules on wastes banned from the trash.
CalRecycle says it's illegal to put hazardous waste in the garbage. That covers latex and oil-based paint, solvents, antifreeze, batteries, motor oil and filters. Tires aren't hazardous, but they're banned from the trash too. Treated wood with chemical preservatives has to be handled as hazardous waste, and so do propane tanks and other compressed gas cylinders.
Beyond those, hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster. Older roofing, flooring and siding can contain asbestos, so have it checked before a tear-out.
Because Palm Desert projects must meet a recycling target, clean loads of concrete, metal or green waste may be handled apart from mixed trash. If you're unsure about an item, ask your account manager before you load it.