Running a generator out of gasoline is not gonna hurt the generator or the motor.
The only thing that will happen to the generator is that the electricity starts to flicker as the engine revs up and down while the gasoline is leaving the carburetor.
Then it will eventually just shut off and not run again until you add more gasoline and start it back up.
When you put a generator into storage you should let the engine run until the carburetor runs completely out of gasoline to keep the gasoline from turning to varnish in the carburetor.
I've run my generator out of gasoline completely a lot for many years and it never has harmed the generator or motor.
Just like running a car out of gasoline it's not gonna harm it either and you just have to put more gasoline in it for it to run again.