The symbolism in this story relates primarily to the double meaning of "Phoenix" that is alluded to in the title. The Phoenix is of course not only used to refer to a city in the state of Arizona but it is also a mythological bird in Egyptian mythology that periodically dies and then is reborn in its own ashes to rise again. It was used by the Egyptians as a powerful symbol of regeneration and immortality. In this short story it is used to symbolise the act of reclaiming that which has been lost.
If we examine the story carefully, we can see that this not only refers to the ashes of Victor's dead father, but also the ashes of Victor's life and the way that he has chosen to reject his Indian roots and turn his back on his cultural heritage. As his name suggests, it is Thomas Builds-the-Fire who metaphorically sets alight the fire under Victor in order to help this regeneration occur. The symbolism of the title therefore points towards one of the central themes of the story, which is the way in which we can both turn our backs on our cultural heritage but also be reborn to them. This is something that Victor, by the end of the story, makes it clear that he is ready to do when he promises he will listen to one of Thomas's stories.