Dumbledore was on his feet again, pale as any of the surrounding Inferi, but taller than any too, the fire dancing in his eyes; his wand was raised like a torch and from its tip emanated the flames, like a vast lasso, encircling them all with warmth.
Then he heard a terrible cry that pulled at his insides, that expressed agony of a kind that neither flame nor curse could cause, and he stood up, swaying, more frightened than he had been that day, more frightened, perhaps, than he had been in his whole life...
Which came first, the phoenix or the flame?
Like a tongue on frozen steel, like flesh in flame —
Harry was just thinking that all he needed was for Dumbledore's pet bird to die while he was all alone in the office with it, when the bird burst into flames.
Fawkes is a phoenix, Harry. Phoenixes burst into flame when it is time for them to die and are reborn from the ashes.