Why didn't Fawkes rescue Dumbledore in the Half Blood Prince?

Was Dumbledore simply prepared to die that night?

Yes.

In fact, as is explained in Deathly Hallows,

Yes

Dumbledore was injured while playing with the Black Stone. He was getting weaker which was observed when he fought with Voldemort in OOTP. If he had continued living, there was a chance that Voldemort might defeat him and become the lawful owner of the Elder Wand.

He chose to die instead at the hands of Snape by choice so that the ownership of Elder Wand stays with him and it can not be used to cause more harm in future. Hence the Fawkes didn't come to help him.

I'm rather surprised, the most obvious answer here seems to be missing.

How?

How could Fawkes have helped him? A phoenix is a powerful magical creature, yes, but it has shown no battle prowess beyond that of a strong bird. Fawkes hurt the basilisk, sure, but only by attacking its eyes, which doesn't require any great strength. It has no offensive magical abilities that we've seen. A bird versus a bunch of armed Death Eaters - not exactly a fair fight.

It could show up and swallow the Avada Kedavra from Snape, sure (though of course by that point Dumbledore wouldn't want it to) but then what? You'd be back in the same position, save with a baby Fawkes (that the Death Eaters could perhaps kill for real, somehow? Maybe I should ask a question about phoenix mortality...). Dumbledore would still be nearly dead and wandless, surrounded by Death Eaters.

With or without Fawkes, Dumbledore's fate was sealed the moment he immobilised Harry.

AMENDMENT

It's been pointed out to me that Fawkes could have teleported Dumbledore away. I suppose this is true, though, in canon, we have no explanation of how and why Fawkes appears when he does. It's likely he has some form of autonomy, since he appears to Harry in book 2 (though at a stretch maybe you could claim Dumbledore had some sort of "help-student-who-needs-it" spell going on). He's with Dumbledore when he flees in book 5, and then appears during the duel (again, it's not clear if he appeared of his own will or was summoned, though Dumbledore's calm demeanour suggests summoned to me).

Not much info to go on! Perhaps it takes energy/strength for Dumbledore to summon him that he simply didn't have in the tower?

Perhaps the potion blocked the connection between the two? That would make sense, since Voldemort didn't want anyone being rescued from the cave.

Unfortunately, without any more canon info, it's largely just speculation. Hell, perhaps Fawkes was out on some Order business, or even just sleeping!

One slightly more informed possibility is that Dumbledore judged the risk of leaving an immobilised Harry - even an invisible one - with the Death Eaters too high. After all, if one were to trip over him, he'd be dead. Or I could absolutely see a scene where Greyback stops mid-sentence and starts sniffing the air, moving towards Harry... (Do werewolves have heightened senses in the HP world?)

Fawkes may be just temporary bonded to Dumbledore as he is an effective immortal being. So Fawkes was not Dumbledores pet but his friend. A reason why Fawkes didn't save Dumbledore can be that it was fate/destiny that Dumbledore had to die, that his life is over. Also there is this curse on the Gaunt Ring/Resurection Stone which allowed him just one year to life on - and only because of Severus Snapes' quick intervention. Even Fawkes may not been able to save him and as a creature of light he was compulsed to allow Dumbledore to die without pain.

Yes, I think Dumblerdore was ready to die, as soon as the "time would be appropriate":

• because he was dying anyway, because of the ring curse
• and he had planned to use this and to choose the way he died in order to:
  • kill the power of the Elder Wand
  • increase Snape's credibility to Voldemort
  • "Save" Draco

Of course, all this was planned since a few months, and, if he did not choose to die earlier, why that night, why not try and gain a little bit more time?

My guess is that :

1. His remaining life expectancy had lessened hugely that night. Like he said, the potion he drank in the cave was no health drink. He was awfully weakened, even struggling to stand. No certainty, but he would probably have died anyhow a few hours or days later because of that potion.

2. It's never really precised how the Unbreakable Vow time limit functions, but the situation described in Snape's vow to Narcissa was clearly happening : Draco was in the position to kill Dumbledore, and yet seemed unable to do it. Snape was there and aware of it. I think not fulfilling his vow at this moment would have meant breaking it. So the situation itself implied that Dumbledore and Snape's plan had to happen at that moment, and they both knew it.

So, maybe, probably or certainly, Dumbledore could have fought and defend himself, with different means, even despite his weakness. But he didn't wanted to, he understood the time for his plan realization was there.
(I also think this is the reason why he immobilized Harry.)

And Fawkes, who respected Dumbledore, respected his plan and his will.

I think that Dumbledore just told Fawkes not to save him because if Snape hadn’t killed him that night either the Unbreakable Vow would have taken into effect or Voldemort would have killed him and Dumbledore needed Snape in Voldie’s good books.