I know that temperature plays a crucial role in the process of ignition, as most combustible materials will spontaneously start burning in presence of enough oxygen when heated above the kindling point. Anyway, I'm not really clear whether or not heat is just as important to keep a fire alive when a material is already burning, in which case, I suspect, other factors may contribute to balance the chain reaction. If this is the case, would it be possible to extinguish a fire by cooling the fuel surface below a certain temperature? Could you for example put out a pan of burning oil by pouring more very cold oil into the pan, or would that just make the fire last longer?
Generally speaking, solids and liquids don't burn. They get hot, liquify (if solid), vaporize, and then the vapors burn.
So, if you cool the solid/liquid somehow, the energy that was vaporizing the fuel now must heat up the fuel and then vaporize it. If you cooled it enough, it could require more heat to continue vaporizing the fuel than is available from the flame.
Now, in practical terms, I would not try to put out a grease fire with more oil. That just sounds like a recipe for disaster.
George Goble won the 1996 Ig Nobel Prize in Chemistry by pouring liquid oxygen on a barbeque containing unlit charcoal briquettes and a smoldering cigarette. It did not put out the cigarette. It melted the barbeque.
If he had soaked the briquettes in liquid oxygen first, it would have been the equivalent of lighting the barbeque with sticks of dynamite.
Things burn a lot better in pure O2 than in air. In air, N2 absorbs some of the heat of combustion without contributing any.
Yes it is possible to cool a fuel to a point where it won't burn in air. See http://en.wikipedia.org/wiki/Flash_point
For a list of examples, see http://en.wikipedia.org/wiki/Flammability_limit