Why would my water heater start running if my water is shut off

Bypass valve is to bypass the water heater. No bypass valve and always water in tank. Put a bypass valve and shut it up it is like no bypass valve.Open the bypass and shut up the input (cold water) and output hot water) no running water in heater tank but still have cold water in the rest of the house even in the hot side of the faucet.

You have run out of hot water. Read up on how hot waters work and the different types.
Standard Electric Operation

Energy Smart Electric Operation

How Whirlpool Natural Gas and Liquid Propane Water Heaters Work

How It Works Water Heater

It has to cool down for safety reason unit still to hot

your tank is probably going bad in your water heater the shut off valve would be your homes water supply valve.

1) Look at the top of the old tank.
There will be markings on the tank, probably embossed in the steel cover that show cold - hot.

2) Many tanks have red and blue plastic pieces surrounding the inlet pipes

3) If there is a shut off valve located above tank, it should be on cold water line assuming house was plumbed to code.

4) When we install water heaters in old houses where water heater is removed, we look for shut off valve above tank that tells us which line is cold. If tank is missing and pipes have no shut off valve, we turn on outdoor water supply and see which of the two water heater pipes starts running water first ... because that is cold side.

5) For do-it-yourself water heater information about electric water heaters:
http://waterheatertimer.org/DIY.html

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I have a code showing C7 13 the heater runs for about 2 to 3 mins then shuts off. Any suggestions?

You may have a control board issue, there is no sensor that is IN the water tank your thermostat controller are exterior to the tank under the black foam looking piece which I think is where your trouble lies

water heaters run until they reach a certain temperature and then shut off, in a similar way to the thermostat in your air conditioning.

if you have the exhaust run together it will not light up,, they have to be seperate,, the computer board sences the unburnt gasses and will not allow it to start,,