Turn off water supply but leaving heat burner on?

With the water turned off to standing pilot gas water heaters, I suggest leaving the controller in pilot; that way it can not turn on the burner and damage the tank. This also keeps the pilot running and those pesky little spiders that love to build nests in the tubing away; then, when you return, just turn it to the proper temp: no hassle having to relight the pilot. But you need to drain the pipes as no heat and water in the pipes can damage them.

If you have electronic ignition turn the control to off.

Since this is a recirculating heating system, though, I would suggest leaving the water in the system and turning it down. That will keep the pipes warm so breakage is not a problem unless your vacation is extended (a month or more, then draining makes more sense).