The monitor shouldn't care. The DC power supply you may have installed for a dedicated game board might not like running with no load, but most seem to just shut down safely.
What may be happening is that the PC may be able to "detect" the monitor connected when it's off (since the inputs may be fairly low-Z when there's no power) but not when it's on (as the inputs may go fairly high-Z relative to the 75ohms the PC expects). Sounds odd, I know. If it doesn't detect a connected monitor, it may run in a different than intended "default" mode. Try either "forcing" that the monitor is connected on the appropriate output, if your driver allows that, or strap some ~75-100 ohm resistors between the RGB signals and ground at the input to the monitor (you may need to turn up contrast or get a video amp, then, unfortunately).
What model monitor is this? Is it a multisync? Does it have a PC style HD15 input for video or just something more "arcadey"?