What causes the fuel pump to run with the engine off and key out?

Common for GM in tank fuel pumps to continue to run for a few seconds after engine is switched off. IF pump is *continually* running with key off - then a problem exist - possibly fuel pump relay.

Just to be sure you selected the right relay - the pump should not have turned with the relay removed .... Suspect a wiring problem - however that will require *hands-on* testing. Good luck!