Hat tip to Richard Packham for these quotes.

Prophet Joseph Smith:

Says one, “I cannot believe in the salvation of beasts.” Any man who would tell you that this could not be, would tell you that the revelations are not true. John heard the words of the beasts giving glory to God, and understood them. God who made the beasts could understand every language spoken by them. The four beasts were four of the most noble animals that had filled the measure of their creation, and had been saved from other worlds, because they were perfect: they were like angels in their sphere. We are not told where they came from, and I do not know; but they were seen and heard by John praising and glorifying God (See Rev.4:6). – General Conference held on the floor of the Nauvoo Temple, April 8, 1843 HC 5:343-44, cited in McConkie, Mormon Doctrine 1st ed p 578, and in Teachings of the Prophet Joseph Smith 345-346

Prophet Joseph Fielding Smith:

Animals do have spirits and that through the redemption made by our Savior they will come forth in the resurrection, to enjoy the blessing of immortal life. – “Answers to Gospel Questions” Volume 2, Page 48

Apostle Bruce R. McConkie:

Nothing is more absolutely universal than the resurrection. Every living thing and being will be resurrected. “As in Adam all die, even so in Christ shall all be made alive” (1 Cor. 15:22). … Just as the creative and redemptive powers of Christ extend to the earth and all things thereon, as also to the infinite expanse of worlds in immensity, so the power of the resurrection is universal in scope. Man, the earth, and all life thereon will come forth in the resurrection. And the resurrection applies to and is going on in other worlds and other galaxies.

Thus saith the Lord: “And the end shall come, and the heaven and the earth shall be consumed and pass away, and there shall be a new heaven and a new earth. For all old things shall pass away, and all things shall become new, even the heaven and the earth, and all the fulness thereof, both men and beasts, the fowls of the air, and the fishes of the sea; And not one hair, neither mote, shall be lost, for it is the workmanship of mine hand” (D. & C. 29:23-25). – Mormon Doctrine 1st ed 573-578, 2nd ed 642-643

The First Presidency (Joseph F. Smith, John R. Winder, Anthon H. Lund):

He made the tadpole and the ape, the lion and the elephant but He did not make them in His own image, nor endow them with Godlike reason and intelligence. Nevertheless, the whole animal creation will be perfected and perpetuated in the Hereafter, each class in its ‘distinct order or sphere,’ and will enjoy ‘eternal felicity.’ That fact has been made plain in this dispensation (D&C 77:3). – Church First Presidency Message, Christmas greetings, Dec. 18, 1909

Apostle Joseph Fielding Smith (later President):

The Lord created all things for a purpose. Nothing has he created to be destroyed, but that all things might endure forever… The Lord intends to save, not only the earth and the heavens, not only man who dwells upon the earth, but all things which he has created. The animals, the fishes of the sea, the fowls of the air, as well as man, are to be re-created, or renewed, through the resurrection, for they too are living souls. – General Conference, October 1928

Apostle Joseph Fielding Smith (later President):

Every creature has a spirit, and that it existed in the spirit before it was on the earth; the spirit of every creature is in the form of its temporal, or mortal, body. Since this is true, and all forms of life partook of the effects of Adam’s fall, therefore they are entitled to the resurrection and shall live again. …

Likewise the earth, which is a living body, must die “in like manner” as to all other mortal things, and then receive the resurrection (Isa. 51:6). The fact that the spirit of every animal, every fish, every fowl of the air, is in the likeness of its body, and that also it was created in the spirit in the beginning, is a contradiction of these unscientific theories which man has inflicted upon a fallen world. – “Church History and Modern Revelation”, Published by The Council of The Twelve Apostles of The Church of Jesus Christ of Latter-Day Saints, 1946.