Defective pressure or flow switch

All hot water pressure washers should have a pressure switch or flow switch for safe operation. Some machines have both a pressure switch and flow switch.

The purpose of these switches is to turn the burner off if water stops flowing through the burner.  That's why the burner shuts down when you release the trigger on the gun.

I these switches malfunction it could be the reason your burner will not fire.  Conversely a malfunction may prevent your burner from shutting down.  Check the continuity of these switches with a multimeter, while turning the feed water on and off or stopping and starting the pump.