This answer is based on the Christian theology of Emanuel Swedenborg (1688-1772), and reflects the views of the "Swedenborgian" or "New Church" denominations that accept his theology. Except for the last part, about the "Peaceable Kingdom" passages in Isaiah, this answer is extracted and edited from my article, "Will We See our Pets Again in Heaven?"
Animals do not have eternal souls
It would certainly be more popular among animal-lovers to say that animals have eternal souls. However, from a Christian perspective, while animals do have souls, their souls are not eternal, as human souls are.
Animals do have souls. Without souls they could not be alive. Our body is not alive by itself. Rather, it is our soul that lives within our body, and gives life to our body. When we die, what actually happens is that our soul leaves our body. Without the soul to give it life, our body is as dead as a doornail. It's just a very complex piece of organic matter that soon decomposes and is reabsorbed into the earth.
Animals, too, must have souls in order to be alive. When they die, their souls, too, leave their bodies. However, unlike human souls, they do not go on to a conscious existence in the spiritual world. Instead, they are reabsorbed into the spiritual ecosystem just as their bodies are reabsorbed into the material ecosystem.
Animal souls vs. human souls
Why the difference between animal souls and human souls?
Animal souls consist only of the lowest level of spiritual reality. This level has to do with taking care of basic physical and psychological needs such as food, water, shelter, reproduction, and the companionship and protection of the group in social animals. Animals need these external, earth-oriented spiritual levels so that they can go about their daily lives seeking and obtaining what they need to continue living and perpetuate their species.
Human souls add the higher, more spiritual and heavenly levels of spiritual reality. These levels give us:
- The ability to think rationally and consider not only different
possible courses of action, but their various moral, ethical, and
- The spiritual freedom to make moral, ethical, and spiritual choices,
such as whether we will act from love and concern for our fellow
human beings or act only for our own advancement and pleasure.
- The ability to think about God and the spiritual world, and make
goals that involve more than satisfying material and social needs. We
humans are able to direct our life according to the will of God as we
understand it, and to act for the eternal wellbeing of ourselves and
our fellow human beings.
These are all things that lower animals simply can't do.
And these are precisely the things that make human souls eternal. In particular, our ability to form a conscious relationship with God is what makes our soul eternal. Even if we choose to reject our relationship with God, the fact that we have the ability to do so means that a human being's soul will never cease to exist as a conscious, self-aware being.
Animals do not have spiritual freedom or rationality, and cannot even think about God, let alone form a relationship with God. Their souls are focused on the things of this earth, not on the things of heaven. That's why their conscious existence is limited to this earth. In a sense, this earth is their heaven.
There are animals in heaven
Just because animals don't have eternal souls, that doesn't mean there are no animals in heaven.
All of the animals we see here on earth also exist in the spiritual world—and many more that don't even exist here!
However, the animals in the spiritual world are not the souls of animals who have died on earth. Instead, they are reflections and expressions of the thoughts and feelings of the (human) spirits and angels who live in the various areas of the spiritual world.
Heaven contains beautiful and gentle animals because they reflect and express the state of mind of the people there. Different angels and communities of angels will have different animals because angels, like people on earth, are different from one another.
For the same reason, hell contains frightful and vicious animals because they reflect the state of mind of the evil spirits there. In fact, from a distance, evil spirits themselves often look like fierce, predatory animals—because that's exactly what they are like in character and personality.
In short, the animals in the spiritual world do not come from the material world. Instead, they come into being in the spiritual world based on the thoughts and feelings of the people there.
"The wolf and the lamb shall feed together"
Swedenborg interpreted Isaiah 65:25 metaphorically, as referring to the peace and protection from evil that will be enjoyed by those who are a part of the Lord's kingdom in heaven.
Here is what he says about a similar passage from Isaiah in Apocalypse Explained #314:3, in a fairly old-fashioned translation—which is all that's in print for this work. I have edited to make it a little more understandable for today's readers:
The wolf shall dwell with the lamb, and the leopard shall lie down with the kid; the calf and the young lion and the fatling together; and a little boy shall lead them. And the she-calf and the bear shall feed; their young ones shall lie down together; the suckling shall play on the hole of the adder, and the weaned child shall put his hand on the cockatrice's den. (Isaiah 11:6-8).
These things are said about the Lord's advent, about His kingdom, and about those in it who are engaged in the good of innocence: that they will have nothing to fear from hell and the evils that come from hell because they are protected by the Lord.
The Lord's kingdom is here described by innocence of various kinds, and by their opposites, from which they will be defended. The lamb means innocence of the inmost level; its opposite is the wolf. The kid means innocence of the second level, of which the opposite is the leopard. The calf means innocence of the lowest level, of which the opposite is the young lion.
Innocence of the inmost level is the kind that belongs to those who are in the third or inmost heaven, and its good is called heavenly good. Innocence of the second level is the kind that belongs to those who are in the second or middle heaven, and its good is called spiritual good. And innocence of the lowest level is the kind that belongs to those who are in the first or lowest heaven, and its good is called natural-spiritual good.
Because the goods of innocence are described by these, it is therefore also said, "And a little boy shall lead them," and also, "The suckling shall play on the hole of the adder, and the weaned child shall put his hand upon the cockatrice's den." A boy, a suckling, and a weaned child also symbolize those levels of innocence.
You can see the original translation from which I edited this here. Scroll down to the paragraph that starts with a 3 in brackets.