Stop in for a cup of coffee

Does it have an actual pilot light that is burning all the time? Is that what goes out?

If so, it is that tube that contains the jet inside. You can usually disconnect it (after turning the gas off) and blow it out backwards with compressed air. Sometimes you need to run a wire or needle down it.
It has a pilot light. I might have to give that a try tomorrow. You can manually light it and the burner works and shuts off and the pilot goes out.