The Honda EU2000i has a small fuel pump and can suck the fuel out of an external tank. In order to do so you will need to modify your existing fuel cap or buy one already modified.
The basic premise is you remove the vent from the center of the cap and replace it with a barbed fitting. Then put a fuel safe hose on the barbed fitting and run it over to your extended run tank.
As far as refueling while the generator is on, well that's up to you with how you hook to the extended tank and possibly how far away the tank is from the generator.
there are replacement caps available on Amazon. But the reviews aren't all that good. I like this youtube video as it shows how to use your existing cap and also includes all the parts in the description.
Link