If I've already unlocked the upgrades provided by an HQ, is there any reason to keep it on? Can I just shut it off and save on the maintenance and utilities it requires?
If you turn it off you will lose the ability to use your special buildings. This will also happen if you are playing in a shared map, and someone turns off a shared resource. This happens, for instance, when you have your City Hall built up in a region with say Department of Finance. If it is turned off, you lose the ability to tax by wealth, and get bonds over 25k.