First, you're introducing deployment-time dependencies on network resources you don't control. The worst time to discover just how bad an idea that is is when you've just had a critical server go pop while rvm.io (for instance) is having DNS problems, and you've got clients, bosses and clients' bosses breathing down your neck to fix it.

Second, it implies you've got build tools installed on your production machines. That's bad for security.

Third, I've seen RVM screw up too much to trust it. It's got far too many moving parts, and so far I haven't found anything I need which it does that isn't done better by some other simpler tool.