Which reminds me of this quote "It is, of course, extremely toxic, but that’s the least of the problem. It is hypergolic with every known fuel, and so rapidly hypergolic that no ignition delay has ever been measured. It is also hypergolic with such things as cloth, wood, and test engineers, not to mention asbestos, sand, and water-with which it reacts explosively. It can be kept in some of the ordinary structural metals-steel, copper, aluminium, etc.-because of the formation of a thin film of insoluble metal fluoride which protects the bulk of the metal, just as the invisible coat of oxide on aluminium keeps it from burning up in the atmosphere. If, however, this coat is melted or scrubbed off, and has no chance to reform, the operator is confronted with the problem of coping with a metal-fluorine fire. For dealing with this situation, I have always recommended a good pair of running shoes. "

Gravity powered fusion bomb

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.

I can't find anything that says it's not considered fire if it's not burning with oxygen, just that a fuel needs to be oxidizing and combusting, and the definition of oxidation has expanded so it doesn't necessarily require actual oxygen anymore (even though that's how the word originated).

Here's a neat clip I found of hydrogen-chlorine fire.