We know that Voldemort's wand was made from one of Fawkes' feathers, like Harry's. We know that this fact caused the weird behavior of the wands when Harry and Voldemort battled in Little Hangleton Graveyard.
Why could Voldemort successfully use the Killing Curse against Fawkes? Does this kind of magic just not work when the target of the spell is the living provider of the wand's core (instead of a 'brother wand'), or was there ever a more satisfactory explanation to this?
Fawkes himself, as a living phoenix, would be inherently different than either the Dark Lord’s or Harry’s wand with one of Fawkes’s feathers. The twin-core effect seems to only come into play when two wizards with wands with a core from the exact same source use those wands against each other. It seems to be specifically about wands, and happens when two ‘brother wands’ meet.
“So what happens when a wand meets its brother?’ said Sirius.
‘They will not work properly against each other,’ said Dumbledore. ‘If, however, the owners of the wands force the wands to do battle … a very rare effect will take place. One of the wands will force the other to regurgitate spells it has performed - in reverse. The most recent first... and then those which preceded it...”
- Harry Potter and the Goblet of Fire, Chapter 36 (The Parting of the Ways)
Fawkes, as the phoenix that gave the feather, doesn’t necessarily have the ability to do the same thing to wands made from his feathers, since he’s a living creature, not a wand. In addition, it only happens when the ‘brother wands’ battle each other, meaning they’re used against each other in a duel. They can’t just both be present, where either one or both isn’t being used at the time, they have to be actively used against each other to cause it. The Priori Incantatem only happened when Harry and the Dark Lord attempted to duel using their twin-core wands. When they both cast spells at the same time, their two spells meet, and that’s what started the Priori Incantatem.
“Voldemort was ready. As Harry shouted ‘Expelliarmus!’, Voldemort cried, ‘Avada Kedavra!’
A jet of green light issued from Voldemort’s wand just as a jet of red light blasted from Harry’s – they met in mid-air – and suddenly, Harry’s wand was vibrating as though an electric charge was surging through it; his hand had seized up around it; he couldn’t have released it if he’d wanted to – and a narrow beam of light was now connecting the two wands, neither red nor green, but bright, deep gold – and Harry, following the beam with his astonished gaze, saw that Voldemort’s long white fingers, too, were gripping a wand that was shaking and vibrating.”
- Harry Potter and the Goblet of Fire, Chapter 34 (Priori Incantatem)
While both Harry’s and the Dark Lord’s wands were present at the graveyard for the entire encounter, the Priori Incantatem only happened when they attempt to duel and their two spells meet. Neither Fawkes nor his feathers were attempting to duel the Dark Lord. Fawkes wasn’t using any magic, and wasn’t doing anything that could meet the Dark Lord’s spell the way Harry’s did. Therefore, the Dark Lord casting a Killing Curse at Fawkes shouldn’t cause this same effect.
Another thing that’s important to consider is that Fawkes intentionally blocked the Killing Curse. Priori Incantatem happens when two brother wands meet in a duel - but Fawkes wasn’t trying to fight, doing anything to resist, or casting any counter-spell, he was just blocking the curse. Also, he swallows the Killing Curse - it never actually touched any of his feathers.
“But even as he shouted, another jet of green light flew at Dumbledore from Voldemort’s wand and the snake struck –
Fawkes swooped down in front of Dumbledore, opened his beak wide and swallowed the jet of green light whole: he burst into flame and fell to the floor, small, wrinkled and flightless.”
- Harry Potter and the Order of the Phoenix, Chapter 36 (The Only One He Ever Feared)
Even if (and this is still not certain), Fawkes had the ability to cause Priori Incantatem, in this case he had every intention of the Killing Curse hitting him so it wouldn’t hit Dumbledore instead. He wasn’t trying to repel it in the way that would be necessary to cause Priori Incantatem.
The fact that Harry's wand shares a core with Voldemort's does not make him invincible to Voldemort. This is clearly evidenced by the fact that in the graveyard Voldemort was able to perform several curses on Harry without anything out of the ordinary happening.
Voldemort moved slowly forward and turned to face Harry. He raised his wand.
“Crucio!”
It was pain beyond anything Harry had ever experienced; his very bones were on fire; his head was surely splitting along his scar; his eyes were rolling madly in his head; he wanted it to end ... to black out ...to die ...
“And now — we duel.”
Voldemort raised his wand, and before Harry could do anything to defend himself, before he could even move, he had been hit again by the Cruciatus Curse. The pain was so intense, so all-consuming, that he no longer knew where he was. ... White-hot knives were piercing every inch of his skin, his head was surely going to burst with pain, he was screaming more loudly than he’d ever screamed in his life —
“I asked you whether you want me to do that again,” said Voldemort softly. “Answer me! Imperio"
“I said, bow,” Voldemort said, raising his wand — and Harry felt his spine curve as though a huge, invisible hand were bending him ruthlessly forward, and the Death Eaters laughed harder than ever.
“You won’t?” said Voldemort quietly, and the Death Eaters were not laughing now. “You won’t say no? Harry, obedience is a virtue I need to teach you before you die. ... Perhaps another little dose of pain?”
Voldemort raised his wand, but this time Harry was ready; with the reflexes born of his Quidditch training, he flung himself sideways onto the ground; he rolled behind the marble headstone of Voldemort’s father, and he heard it crack as the curse missed him.
The extraordinary things only begin happening once spells generated from the two wands collide with each other. The connection created by the collided spells generated from twin wands is what protects Harry.
This being the case, there would be no reason why Voldemort would not be able to kill Fawkes. When casting a curse at Fawkes there is no other spell that is colliding with Voldemort's spell, so nothing out of the ordinary would happen. In much the same way as Voldemort could (and did) curse Harry to his heart's content if Harry didn't cast any spells back (and presumably Voldemort could even cast spells at Harry's wand, and possibly destroy it) he can curse Fawkes. The fact that Voldemort's wand has a feather from Fawkes is of no import because the extraordinary things are only generated by a collision of spells, something that Fawkes cannot make happen.
