Why won't my gas dryer stay lit?
This could be an issue with the thermocouple. A thermocouple is used to sense whether the gas was ignited by the glow coil and is burning. If it isn't burning, and the thremocouple isn't heated up by the flame, the gas valve is shut off to prevent excess gas buildup.

Although this might not be the exact problem, it might lead you down the right path to find it.