Among other things, fire represents light, the light at the end of the tunnel which the stranded boys on the island eventually hope to reach. So long as the fire can be kept going, there will be hope that the boys will one day be rescued. But once it goes out, that hope will instantly die out with it.
It's notable in this regard that the very boys charged with looking after the fire—Jack and his gang—couldn't care less about its going out. They'd much rather head off for another pig-sticking adventure or bully some littluns just for the hell of it.
Jack realizes early on that it's not in his interests for the other boys to have hope or to see light at the end of the tunnel in the shape of a possible rescue. He knows that, as long as the boys remain stranded on the island, he'll get to be king of the castle. He enjoys life on the island, and in truth, doesn't want his captivity to end. That being the case, he's completely indifferent to whether or not the fire goes out. His neglect of the fire shows him to be the enemy of hope, civilization, and enlightenment, indeed all the many things that the fire symbolizes.