Do animals have spirits? What happens to them after they die?

Yes, animals have spirits (see D&C 77:2–3). Of course, there is a major difference between animals’ spirits and our spirits—we are begotten sons and daughters of Heavenly Father, and they are not.

And according to the Prophet Joseph Smith, there are at least some animals in heaven. He said:

“John saw curious looking beasts in heaven; … actually there, giving glory to God. … (See Rev. 5:13.) …

“I suppose John saw beings there of a thousand forms, that had been saved from ten thousand times ten thousand earths like this,—strange beasts of which we have no conception: all might be seen in heaven. John learned that God glorified Himself by saving all that His hands had made, whether beasts, fowls, fishes or men; and He will glorify Himself with them” (in History of the Church, 5:343).

So, although we don’t have a complete understanding of what happens to animals after they die, we believe that they will enjoy some kind of salvation and immortality.