Burning with oxygen is oxidation, things oxidize when the element oxygen binds with another element, which it’s very prone to doing. Oxidation with iron is rust, it just happens much more slowly (but still exothermically) than when biological compound oxidize they release energy more rapidly, rapidly enough to cause other nearby organic bonds to break and expose themselves to sites for more oxidation to take place.

There can be electrical fires, chemical fires, classic fires, and self fueling fires (the kind you were originally asking about). There are probably more categories now. Always be sure to use the proper fire extinguisher for the fire at hand.