Surburban Water Heater - Not staying lit

You probably have a blockage in the burner FLUE, not the burner tube. The tell is, you left the door open and it appeared to run better. I use a water hose and run water from the TOP(exit) to flood the passage out the bottom where the burner is. Also, use some type of flexible snake after the water is run to knock loose any debris that the water did not wash loose. Mud nests and other insects can build in that curved tube and you cannot see them. Doug

PS You can take it loose(Burner) and roll a golf ball in the exit and see if it comes out the bottom.