The same thing that would happen anytime would happen now.<p>The military isn't suddenly closed down - just the relatively benign and semi-tolerable parts of the government are neutered for political reasons.<p>Getting useless paperwork through the system isn't going to happen today, but you can be rest assured that the IRS will still be collecting taxes and the NSA will still be spying on you.
The same things that would normally happen? Shutdown only applies to non-essential personnel. The only impact to the military are things like the training academies, or recruiting.