Should I turn the hot water tank off or turn it down on extended vacation?
If you have good access to the pipes and can be completely sure that they're empty down to the service entry, then you can get away with turning off the hot water heater and draining the lines/tank.  If it's a newer model, though, they're generally extremely efficient, and won't cost you much to leave on for peace of mind.

If you can't verify empty lines, then no, you probably shouldn't turn off the hot water heater.