2011 Sierra 1500-gas leaking out of tank but only when running.

Sounds like the pressurized gasoline line is corroded or leaking on top of the tank. This would be the line that runs from the pump to the engine. The tank is probably okay because you say it leaks only while the engine is only. Possibly the return line is bad too. I would replace both while the tank is removed. Repair it soon when the gas tank is low on fuel...easier to handle empty tank. Be careful handling gasoline. This happened to me once here in rusty Pennsylvania. Road salt and mud lay on top of the tank.

You might be able to use a mirror to see what's going on with the fuel pump and fuel line on top of the tank to positively ID what is leaking. If you do have to drop tank might want to consider replacing fuel pump just as a precaution in addition to any leaky lines.

Sounds like the gas line, I agree with Hornet_2497 Thanks

I just dropped my tank for the same problem. The smaller branch off the fuel puml/sending unit had rusted off. I cut both ends back to solid material and sliced it with 1/4" fuel hose. Even though the gauge had been at empty for awhile. There. UST gave still been 40 litres in it. Tried standing Ig on end to drain it from the back vent, but nothing came out. What is that blue fitting? I tried to remove it to drain the tank, but was afraid I'd ruin it. It would have been easier to have lifted the box off to make this repair. If not for setting it on fire, I would cut an access panel into the bed.

This is the line that rusted away.