Sounds like when you turn the radio off it's still trying to amplify because you're using the ignition instead of a actual remote turn-on provided by an aftermarket radio. If the amp stays on with no signal to amplify with the gain up, it's going to buzz if it's stupid enough to try to amplify nothing.
Solution 1: Get an aftermarket radio.
Solution 2: Wire a switch to the remote wire so you can interrupt the REM wire.