Paul does say that "the whole creation" (including animals, presumably) suffers because of the fall:
For the creation waits with eager longing for the revealing of the sons of God. For the creation was subjected to futility, not willingly, but because of him who subjected it, in hope that the creation itself will be set free from its bondage to corruption and obtain the freedom of the glory of the children of God. For we know that the whole creation has been groaning together in the pains of childbirth until now.—Romans 8:19-22 (ESV)
Francis of Assisi is famous for preaching to animals. Whether or not this is apocryphal, it illustrates his concern for non-human life. Wikipedia also notes:
Francis preached the teaching of the Catholic Church, that the world was created good and beautiful by God but suffers a need for redemption because of the primordial sin of man. He preached to man and beast the universal ability and duty of all creatures to praise God (a common theme in the Psalms) and the duty of men to protect and enjoy nature as both the stewards of God's creation and as creatures ourselves.
The final words of God in Jonah express compassion for both Nineveh and its cattle.
On the other hand, humanity is clearly distinct from the rest of creation:
Then God said, “Let us make man in our image, after our likeness. And let them have dominion over the fish of the sea and over the birds of the heavens and over the livestock and over all the earth and over every creeping thing that creeps on the earth.”
So God created man in his own image,
in the image of God he created him;
male and female he created them.
—Genesis 1:26-27 (ESV)
It would be a mistake equate the effects of sin on animals with their effect on humanity. It's also clear that creation will be relieved of its curse when we take on our rightful place as children of God.
The Bible, early Christians and many doctrinal frameworks assert creation has suffered as a result of original sin. But I don't know of any Christian group that puts the effects of the curse on animals at the same level as the curse upon humanity.