You can boot it up without the drive connected AS LONG AS you are not connected to the Internet. If you log onto Live with the drive disconnected Microsoft will ban your console for "modding". If you are disconnected, you can have the drive off and nothing will happen. If it's acting up, try unplugging and plugging the drive back in, both the power and SATA cables that go to it.
Also make sure your disk isn't scratched up. If your 360 is an older model, and considering it had RROD, it probably is, then it may scratch disks if you move the DVD drive when it's running (you'll hear a loud scratching grinding noise from the drive and find rings of scratches around your game disks, mine did that when I was testing it, had it all torn apart). My copy of Guitar Hero 3 does that "Put this disc in an Xbox 360 console to play it" thing, what that means is that the 360 isn't identifying the disk correctly, thinking it is a DVD-video disc, and then playing it as one (try putting the game in a DVD player, it does the same thing.
Try cleaning your disc and use a scratch repair kit if you have to. Then just try putting the disc in and out a bunch of times until it works (that's what I had to do, some reason it works fine now though).