According to ancient legend, the phoenix is a bird that lives a long time and dies by bursting into flame. It then is reborn from among the ashes of the fire.
The phoenix is symbolic in this novel because, it is hoped, this is what will happen to the society in the book. It has been killing itself with fire (using fire to destroy all the books) and then it is finally destroyed in a huge flash of fire (the nuclear war).
But because Montag and the other "book covers" still exist, there is hope that, out of the fire, civilization can be reborn.
There are some who believe that the Phoenix is a self-inflammatory migrant bird that keeps killing and resurrecting itself; others believe that the Phoenix represents immortality in that whenever the last Phoenix dies, another arises to take its place. The idea is that only one Phoenix can live at a time; so when it does re-surface from its own ashes, its destiny is to live for at least 500 years before another can come in its place. In Fahrenheit 451, the Phoenix is a simultaneously combustible creature that self-implodes and recreates itself at the same time. Books burn at 450-degrees, and firemen represent a finality in that burning...the flames being the very existence of a destructive force of nature itself. That force of nature mathematically divides and simultaneously multiplies, both destroying and conquering itself all at once. In either case, something must be destroyed in order for something else to live, so perhaps in the burning of books, the destruction of free speech is what gives free speech its life and immortality. By the same token, the naturally "bigoted" nature of man is possibly what is needed to make bigotry a thing of the past, particularly by the hand of those who choose to think in terms of the present and progress over and above the superstitious and mistaken errors of the past. In a strange twist of fate, we can understand that without the end, there is no beginning.