Should the "Instant Hot Water Recirculating System" model #500800 be turned off if you go on vacation and turn off the water supply to the house?

It doesn't have to but if you're turning off the water supply then why not turn off the pump and put the hot water heater down to pilot position or vacation position. There is a selection switch on the timer for on, off, timer contol.

I don't think it has to be turned off but I did when on vacation this week. Why use the power when not needed?

My manual doesn't say whether you have to stop the pump when you turn off the water supply, but I think it would be wise to do so. My pump is plugged into a portable timer because my circulator is mounted in a hard-to-reach location. I've set the pump's built-in timer to be on all the time and programmed the portable timer with the on/off times. For vacations, I'll unplug the pump from the portable timer.