Dumbledore was too noble and self-sacrificial to consider it.
Dumbledore considered the potion to be highly dangerous and wouldn't have considered allowing any other person or creature to suffer in his place by drinking it. He shrugs off Harry's petition on this basis.
"Why can't I drink the potion instead?" asked Harry desperately.
"Because I am much older, much cleverer, and much less valuable," said Dumbledore. "Once and for all, Harry, do I have your word that you will do all in your power to make me keep drinking?"
(Half-Blood Prince, Chapter 26, The Cave).
Conjured birds may not be as valuable as Harry is, granted. But it seems that Dumbledore also thought that he was the more capable person to face whatever Voldemort had in store. That argument would apply to animals as well.
It would be near-enough impossible, as EJS says, to find an animal that would willingly consent to drinking the potion (even one conjured from thin air). The only way to force it to drink the potion would be by using the Imperius Curse, which Dumbledore was not at all down with.
Again, Dumbledore could have forced an elf or some other creature to drink the potion in his place. But Dumbledore is inherently noble and self-sacrificial and wouldn't have stood for such shenanigans.
"You flatter me," said Dumbledore calmly. "Voldemort had powers I will never have."
"Only because you're too - well - noble to use them."
(Philosopher's Stone, Chapter 1, The Boy Who Lived).
Besides, it's not altogether clear that using conjured beasts would be effective. Dumbledore only considered drinking the potion when every other avenue was closed to him.
"This potion cannot be penetrated by hand, Vanished, parted, scooped up, siphoned away, nor can it be Transfigured, Charmed or otherwise made to change its nature."
Almost absent-mindedly, Dumbledore raised his wand again, twirled it once in midair and then caught the crystal goblet that he had conjured out of nowhere.
"I can only conclude that this potion is supposed to be drunk."
(Half-Blood Prince, Chapter 26, The Cave).
Voldemort had successfully closed all the other loopholes. I doubt that he would leave this one open. I'm sure that if you'd conjured an animal to drink the potion for you, or a fish to swim in and bring it up to you, that that wouldn't have worked. Remember that Voldemort took a great deal of care and devotion in protecting his horcruxes.
That's not to say that a 'real' creature like Fawkes couldn't have done it. But, as per my first point, Dumbledore wouldn't have permitted another creature to suffer in his place.