In this case, fire obeys that old lit rule of thumb: if an object or thing appears in the title of a story, it's probably symbolic in some way. The presence of fire in this story represents life, and the absence of it shows that life is running out, as is the case when the man's fire gets blotted out by falling snow and he feels "as though he [has] just heard his own sentence of death" (24). A human needs fire in the arctic in the same way that one needs oxygen underwater. If it runs out, you're pretty much done for.
The man's struggle is a struggle against nature in which fire is the only natural thing in his corner. As the man puts it, "When it is seventy-five below zero, a man must not fail in his first attempt to build a fire—that is, if his feet are wet" (19). In an everyday situation, getting your feet wet might not seem like the end of the world. But when that happens in the frozen Yukon, it's not only building a fire that is important, but building one quickly. Whenever fire comes up in this story, you can always hear a tick-tock, tick-tock in the background—particularly when that fire gets snuffed.
Fire is also important because it highlights the difference between the man and his wolf dog. While the dog "yearn[s] back toward the fire," the man foolishly decides to keep moving toward his goal because he is "ignorant of cold, of real cold, of cold one hundred and seven degrees below freezing point" (16). In addition, the fire symbolizes the comfort the man hopes to achieve by making it to "the boys" at the camp, where "a fire [will] be going, and a hot supper [will] be ready" (4). The dog also yearns for this comfort, but doesn't think they need to go all the way to the camp to get it. "Just take out your matches and make one now!" it wants to shout at the man.