Water heater leaks when using washing machine

Why would a water heater leak only when using a washing machine?

I assume your water heater is leaking at the TP valve and water is drizzling or dripping out of overflow pipe and coming out at base of water heater. If tank is leaking elsewhere, it may be rusted and ready to rupture with each high-heat event. Be aware that tank rupture causes massive damage, and water runs until someone turns off valve located on cold water line above tank.

If your TP valve is leaking occasionally, then riase and then gently lower TP valve lever to clear possible debris.

Next best suspect is high pressure. Use a pressure gauge on drain valve of heater, and on outdoor spigot. Anything over 80 psi can cause heater to expell water at TP valve.
You need a pressure reducing valve and expansion tank.
You can try just the expansion tank and see if it solves problem. Expansion tank is designed to stop TP valve from drizzling and dripping water during high-heat events. Expansion tank is about size of soccer ball or football and 'tees' into the 3/4" cold water line anywhere before line enters tank. Must be on cold water side.
Some municipalities require expansion tanks. Note that expansion tanks also help prevent catastrophic tank rupture as tank gets older.

If high pressure is not the issue, then you can replace TP valve with same temperature-pressure rating. Standard residential TP valves are rated for 100,000 BTU. Hardware store sells them. Be sure to use 6 wraps of teflon tape or pipe joint compound on threads when installing so new valve doesn't leak. Remember to twist new valve very tight into tank so water heater pressure doesn't force water out around thread.

Leaking TP valve can signal a bad thermostat. Or it can mean thermostat is set too high. Check thermostat temperature setting. If setting is 150-135, this is considered high.

For gas heater; Dial down the temperature a bit and see if problem is solved.

For electric heater: Turn off power to water heater, remove cover(s), remove insulation to reveal element and thermostat. The thermostat is rectangular in shape. Use small screwdriver to dial temperature to 125-130. Put insulation and covers back in place so cool air doesn't cause thermostat to misread temp. Turn on power and see if problem persists.

If TP still leaks water, then replace the gas valve thermostat, but not before you eliminate high pressure or bad TP valve.

Lastly, if your tank is not leaking > and your TP valve is good except for drizzling > and gas or electric water heater temp must be kept high … then install an expansion tank on incoming cold water line.
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