The AC compressor does nothing more than provide you with Air Conditioning. Unless the clutch on the actual compressor is locked on, (The clutch is what triggers the compressor to turn. Ever notice when you’re using AC and you’re standing next to the vehicle and you hear wind & intermittent clicking? That’s the compressor turning on and off, e.g. providing cool air.) there’s nothing to worry about. Yes, the freon in the system is hazardous to your health and is known to cause cancer in the state of California, like everything else is, lol, but if it’s already drained out, which if you don’t have AC I’m assuming it already is there’s nothing to really worry about.
Here’s worst case scenario, if the compressor is locked up it can cause you to break the belt. This will cause the engine from running altogether. If anyone you know is familiar with mechanics have them do this for you. Release the tension on the belt going to the compressor and try rotating the compressors pulley. If the pulley moves freely, there shouldn’t be anything to worry about. If it does not, you’ll have to put in a bypass system (this being a shorter belt eliminating the compressor from the system with still operating other vital parts that run off of the belt line or I do believe they make pulleys that do nothing more than finish the belt system by placing it inline where the compressor used to be.) & depending on the options based on your particular model would lead you to the correct replacement.
The compressor doesn’t cool the engine. The cooling system does, e.g. hoses, radiator, water pump, ect..
Also, unless the pulley on the compressor is really screwed up I highly doubt it will catch fire either. Fires under the hood are usually only cause by electrical problems and or fuel delivery problems. I doubt the compressor turning on draws that much power to cause an electrical fire, but I’m no electrician.