IIRC, I read that cutting the engine and extinguishing the fire with a steep dive, is just a myth.
If a fire is caused by the fuel tanks being ignited, then it's the fuel that is burning, and cutting the engines would have no real effect in stopping the fire. (maybe it could help slow down the spreading of the fire?)
So the only way to stop the fire is to stop the fuel mixing up with the oxygen to keep the fire going, and conventional diving speeds of WW2 planes just weren't that high enough to cut off the oxygen flow to the fuels. The flames might temporarily simmer down due to high speeds, but as soon as the plane hits level and slows down, the fire will grow again.
However, in other cases, the flames would die out naturally.
Small, local fires caused by a hit to the hydraulics systems would ignite the oil. After all of the oil in the local hydraulics has been burned out, the flame would die out. I've seen a few guncams where a plane has been hit in the wings by a cannon shell, and a small local fire springs up, but dies out quickly - indicating probably the hydraulic oil of the landing gears placed in the wings, have been burned away.
Or HE shells hitting canvas surfaces and setting flames was also common. I've seen some guncams of German planes hitting Hurricanes with cannons, and its canvas was set on fire. Sometimes it's uncontrollable, other times the flame just eats away some of the canvas, and then just dies out - sort of like when if you burn paper, sometimes it would turn into a fireball, and other times it just dies out after a small flame.