What make and model of fridge do you have? Do you want the LED to illuminate when the burner is actually burning, or when the fridge is on in gas mode?
If the fridge has electronic controls it should be possible to add an LED without too much difficulty. For seeing when the burner is burning, wiring the LED (with appropriate series dropping resistor) in parallel with the gas solenoid ought to work. I doubt the extra 10-20 mA would cause any problems for the controls; but just the same, I disclaim any responsibility should it somehow cause problems.