The most common reason for a burner that fires and then shuts off is that the flame sensor thermocouple is not detecting that the burner has lit. The controller circuit bard will shut off the gas supply if no flame is detected within about 30 seconds of ignition. That's a safety feature to prevent LP gas build-up that could result in an explosion.
The flame sensor is an electrode that sticks into the burner tube. In most recent models it is the same device as the igniter. If it is out of position slightly or if the wire connecting it it the circuit board is corroded, it may not sense the flame. It is quite possible that the high energy ignition pulse will make it through the wire but the tiny curent generated by the flame sensor will not. I would remove, clean and reattach the wire at both ends as the first step. Replacing the igniter/sensor is the next step (not expensive or difficult). A bad circuit board is a slim possibility too, but that would be my very last resort.