I'm not mechanic, but it could be a choke not working properly if it runs fine when warm.

"A choke on a lawnmower engine, or any engine with a carburetor, is a device that reduces the airflow through the carburetor. It causes the fuel-air ratio to be richer, that is, more fuel and less air. This helps most carbureted engines to start while cold and before steady fuel flow is established, but the rich mixture reduces the power and efficiency of the engine tremendously. Leaving the choke on will produce hydrocarbon and carbon monoxide emissions and may cause the engine to stall as soon as any load is placed upon it. That is why engines require either manual release of the choke or have an automated device to back it off. Excessive use of the choke will generally result in fouled spark plugs."

 

Choke is part of carburetor. Try carburetor cleaner.

 

Mine is a diesel but I had a similar problem once. It wouldn't stay running unless I manually held the throttle/run /stop thing in the run position several times. it would eventually over-ride I guess but it gave me the idea to replace that switch. I took a lucky guess and replaced the low oil pressure switch. I don't recall the cost exactly but it was fairly cheap. It ran perfectly fine after that. I bought the switch at the Omega Store by the bridge near Lukewright Street if you know where that is. They specialize in generators and small engines so they might be an option for you if you can't get it fixed yourself. But it has to be taken to them, No home service.
Think about that low oil pressure switch.

 

Gasoline deterioration can sometimes be an issue that diesel engines do not have. Gas that is part ethanol is especially bad and has a short shelf life of about 3 months. After that it can turn into varnish. As one poster stated, carbeurator cleaner might help.

 

Something else you might try if you can reach the air intake is to get the generator running and rev it up and close the choke or put your hand over the air inlet to make it pull a hard vacuum then pull your hand away before the engine dies and repeat a couple of times. This will often make trash in the carb go through.

Fuel lines don't just deteriorate on the outside. If this doesn't fix you up, a carb cleaning may be next.

 

I only buy fuel from Shell, Petron or Caltex. I believe the rest is made in the mountains.