Sounds like subs too me. Or something with the input section of the amp.
Trouble shooting
If the amp turns on, positive, negitive, and remote turn on is working.
Disconnect the speaker wires from the amp terminals.
If you have a meter you can do a ohm check on the subs. If you don't have a meter. Take the + and - of one speaker and tap a "D" cell battery across them, the speaker will kick each time you tap the wires if the coil is good.
Also you don't need a sub to test to see if there audio coming out. Any speaker will do for a quick test. Just don't turn it up. Even a cheap fullrange stock speaker or whatever is laying around will do the trick.
If the subs is good, then something in the amp input or the amp it's self is bad.
Also that remote turn on could be connected to the HU + that turns the HU on and off. It should be labled 12+ switch/ign. This is a wire that shows nothing with the key off and goes to + with the car running. It's the wire that makes the HU shut off when you shut the car off. Not that it's causing a problem, just another option, and you don't need to turn on any lights