If I'm leaving on vacation for a week, I shut the gas valve for the water heater completely off. My pilot burns 4 therms(400,000BTU) per month. If I was leaving for a month in the winter, I'd shut off the main water supply and open all the taps including flushing toilets with the water off to try to prevent the pipes don't burst if they freeze.
If I was to turn the heat off, which I would do if I would be out for two of the coldest months or more of winter. I'd turn off the main valve, cut the copper, drain the hot water tank, open all the water valves in the house and blow things dry. Drains will need 'RV antifreeze' added to the traps including the toilets. Resolder the cut section, refill the hot water tank, and turn the heater back on when I get home.
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