Are you certain you are wired into the relay contacts of the opener and not the safety eye or something else? I am not familiar with the Genie openers but all the liftmasters we have integrated have the "smart" openers and we connect directly to the opener and I can stand there for hours and open, pause, and close every door.

Another thing to check is to make sure that you aren't holding the relay closed on your button press. I noticed you posted the logic for your system but I would check continuity on your processor output and ensure that the relay isn't being held.