Hi and welcome to Fixya I am Kelly. This symptom is typical of fuel starvation. It can be caused by
1. A dirty fuel filter (if installed)
2. Debris in the bottom of the fuel tank
3. Fuel shut off selector not all the way on or debris in pet-c^ock bowl. This is located on the bottom of the fuel tank.
4. Float needle assy not allowing proper fuel flow in the float bowl
5. The fuel supply line or the plastic "Y" just before the carb is cracked and air is being introduced into the system as the fuel hose swells from heat.
I really think you have a fuel supply issue. A break down can be found here of your carb:
http://www.boats.net/parts/search/Honda/Generator/0/EU3000IS%20A%20GENERATOR,%20JPN,%20VIN%23%20EZGF-1000001%20TO%20EZGF-1079999/CARBURETOR/parts.html
Fuel tank cap removed
You should be able to remove the angled drain screw (Item 6 above link) in the bottom of the float bowl and have fuel flow freely out of the hole. (I use spray paint can cap / lid to catch the fuel. If you can not fill the spray paint cap to the inner circle of the cap in one continuous attempt there is a problem with fuel supply ( 1 - 5 above )
If you can avoid it DO NOT remove the carb. Getting all the linkages correct during reinstallation is tedious even for seasoned techs on this generator. You can safely remove the float bowl and check for debris / give the jets a blast of carb cleaner. If you do use carb cleaner also direct a shot into the fuel hose connection (hose removed) on the side of the carb with the bowl off. The jet is in the very bottom of the center post.
PLEASE do a leak test before starting the engine with the fuel selector on. (Safety issue)
Make sure you replace the fuel tank cap before starting the generator.
Thanks for choosing FixYa.
Kelly