Eh, I was barking up the wrong tree with the gas cap theory.
It's still doing the same thing. Once I shut it down, it won't draw fuel from the tank to restart. It will suck it out of a gas can no problem, however. If I use a hand vacuum pump to get some fuel in the line from the tank, and then reconnect it to the generator it will fire up and run forever.
I was thinking maybe there's an obstruction in the line, but that wouldn't explain why it continues to run once I manually suck some fuel up to it.
Do these pumps get weak? Mine is an expensive one - looks like $110-$160 depending on the vendor.