Off can't lose money
As far as I know it doesn't take any more energy to heat cold vs hot water a degree of temperature. So consider the energy required to heat the water from room temp to usable hot water as 100 units of energy (arbitrary measure). Say that the standing tank loses heat at 2 units an hour and if you left the tank on it would use two units of energy to replace what was lost.
Suppose you went away for a day and shut the tank off. It would cool down gradually and lose 48 units of energy (I am simplifying here, the loss would not be a steady rate). When you got back it would use 48 units of energy to bring the temp back up again. There is no significant difference between the energy used at 2 per hour or all 48 at once. So, even though there would be no savings there would be no loss either.
Now consider a longer period. At some point the tank would lose all 100 units of energy and assume room temp. At that point it will lose no more energy. If you left it on it would have continued to use 2 units an hour to maintain temp. If it was off, turning it on when you got back would simply require the 100 units to bring it back up to temp.
So in my mind, from purely an energy standpoint, you can't lose by shutting it off. At some point (a couple days?) you will start to save.
Someone mentioned watching the meter spin like crazy when starting up the tank. That's true but it doesn't matter. The meter spins because it is pumping all the 100 units into the tank rapidly BUT the cost of each individual unit is the same. If you are charged 3 cents per unit you are charged the same if the units are pumped in within a 30 minute recovery or they were pumped in gradually over the last two days. The issue is the amount of energy, not the speed at which is is delivered. The price per unit remains steady. Compare it to filling your car's gas tank. Are you charged differently if you slowly fill the tank vs. pumping it as fast as possible? No. The impressive speed of the numbers on the gas pump when pumping rapidly make no difference. All that matters is the final gallon total.
Also there is less wear and tear on the heating elements when you shut it off..
As to the health issues, I can't speak to that. I think that many, many people shut off their hot water and leave water standing in the tank without problems. I look at it as one of the struck by lightening issues of life, possible but very unlikely. If the tank has been standing for a long period of time it's probably not a bad idea to flush it before turning the heat source back on. I would think the same issue applies to the cold water tank so if you are concerned about this you need to flush your whole water system.
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