When the saw is at an idle with the throttle plate closed, there isn't much air flow across the discharge nozzle (check valve) to create a low pressure on the discharge nozzle. That's why the check valve is there, it prevents air from being pulled back into the 'wet side' of the carb.
Inside the carb, the fuel is being sucked out of the 'wet side' of the carb. It is easier to suck in air through the discharge nozzle than fuel through the metering system. Eventually, you run out of fuel and the engine quits idling.
If you have a saw that will idle for a little bit, then have the revs increase a bit (leaning out) just before it dies, that could be a symptom of a bad check valve. If you have to use the choke to get it restarted (even when the saw is already warm), that's another sign it is a bad check valve.
Unfortunately, I don't know of a way to fix the bad check valve other than putting in a new discharge nozzle. However, they're not that expensive and easy to replace.