That effect only happens with twin core wands forced to duel.
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.
In addition, Fawkes didn’t resist, he willingly blocked the curse.
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.