Here's what i would do, Locate the ignition coil, it should just be removing the side blower housing cover. Trace the spark plug wire if you need help locating it... there should be 2 wires on the coil. The spark plug wire and the coil Kill wire.
unplug the coil kill wire. and start the engine. If it dies, you then know it is a problem fuel problem, or fuel solenoid problem. If it Does not die, there is something grounding coil, such as a bad low oil switch, or may even be the coil is failing when it heats up.
Can you start the engine immediately after it dies?