Years ago when the plane (Warrior) was on rental a pilot set it on fire while starting and the aircraft was only saved by another pilot who used his own extinguisher (£300). Damage was limited to scorched hoses, wiring, etc.
My late partner Trevor Shaw was a race car mechanic and he did some tests on the priming options. We knew that the pilot was in the habit of pumping the throttle and we found that only three pumps was enough to have liquid fuel running out of the air changeover box beneath the carburettor. Any blowback on firing would set this off and because the fuel was running over the fibre glass bottom cowling which is rich in paint, oil residue, etc. etc. and fire would take hold quickly.
On the other hand, it took a great deal of priming to get any fuel to run back out of the carb. (I remember 15 strokes being mentioned but please don't rely on that!)
The problem with pumping while cranking is that there probably isn't enough airflow to guarantee that the fuel is drawn into the engine. In my opinion it's only safe to pump the throttle once the engine is actually firing.
Many instructors, and also some engineers that I have discussed this with, appear to be confused about this and continue to advocate throttle pumping. My experience says: NO!. Do not pump the throttle until the engine is firing and only then if you have to to keep it 'alive'.
Underlying all this is the poor starting which besets many rental aircraft and encourages 'innovative' starting techniques. The best insurance against fire is to get the engine fixed so that it starts properly.