Where do animals go when they die?
We don't know. The Bible doesn't say.
Some say when they die they are just dead and have no soul to go anywhere. Others wishfully say they go to heaven. In the Bible, Ecclesiastes 12:7 says about man, "Then shall the dust return to the earth as it was: and the spirit shall return unto God who gave it." Genesis 2:7 says "And the Lord God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul." This was not said about the animals. So perhaps they have no consciousness after death. In that they are better off than most of mankind, who face eternal punishment. But I would rather not be like them, for I can be with God for eternity, as long as I obey Him.
I'm sorry I can not give you a more comforting answer, but the Bible is pretty much silent on this matter.
Note: an observant reader pointed out that the Hebrew word for soul is used for both animals and man. Thus animals are souls. However, the New Testament does make a distinction between the soul (life-force of man and animals) and the spirit which lives on after a man dies. It is this spirit that the Bible is silent about in relation to animals.