Isaiah 11:6-9 raises the very interesting question of whether or not "all dogs/animals go to heaven". We have some indication that animals go to heaven (the word translated "living creatures" in Revelation is the Greek word for animals), but we have every reason to believe that there will be animals in the new creation. We don't know if every single animal will be resurrected in glory, but it seems likely that at least some will.
Scripture shows us that God loves animals (Ps. 104; Job 38-40) and has a habit of sparing animals from His divine wrath against sinners. Animals were spared from the Flood and were taken aboard the Ark with Noah's family. Animals in Israel were to be included in the Sabbath rest of God's people (Ex. 20). Balaam's donkey was spared when it turned aside from the angel (Num. 22). The end of the book of Jonah tells us that God relented from destroying Nineveh, in part, because He was concerned about all the animals in and around the city (Jon. 4).
C.S. Lewis's Chronicles of Narnia are filled with biblically grounded speculation about what animals might be like in the Resurrection world. In his book The Problem of Pain, Lewis gives this helpful reminder:
"The error we must avoid is that of considering them in themselves. Man is to be understood only in his relation to God. The beasts are to be understood only in their relation to man and, through man, to God.
Let us here guard against one of those untransmuted lumps of atheistical thought which often survive in the minds of modern believers. Atheists naturally regard the co-existence of man and the other animals as a mere contingent result of interacting biological facts; and the taming of an animal by a man as a purely arbitrary interference of one species with another. The "real" or "natural" animal to them is the wild one, and the tame animal is an artificial or unnatural thing.
But a Christian must not think so. Man was appointed by God to have dominion over the beasts, and everything a man does to an animal is either a lawful exercise, or a sacrilegious abuse, of an authority by divine right. The tame animal is therefore, in the deepest sense, the only "natural" animal--the only one we see occupying the place it was made to occupy, and it is on the tame animal that we must base all our doctrine of beasts.
Now it will be seen that, in so far as the tame animal has a real self or personality, it owes this almost entirely to its master. If a good sheepdog seems "almost human" that is because a good shepherd has made it so."
Along similar lines, John Wesley, the founder of Methodism, preached a sermon on "The General Deliverance" of the lower creation. He said this about the redemption and restoration of animals as an effect of Christ's saving work:
"But will "the creature," will even the brute creation, always remain in this deplorable condition? God forbid that we should affirm this; yea, or even entertain such a thought! . . . The whole brute creation will then, undoubtedly be restored, not only to the vigour, strength, and swiftness which they had at their creation, but to a far higher degree of each than they ever enjoyed. They will be restored, not only to that measure of understanding which they had in paradise, but to a degree of it as much higher than that. . . . Whatever affections they had in the garden of God, will be restored with vast increase; being exalted and refined in a manner which we ourselves are not now able to comprehend. . . .
I disagree with some of Pastor Randy Alcorn's eschatology, but his book entitled Heaven is filled with solid biblical teaching about heaven and the new creation. If a 450-page book is too daunting, he also has a children's version (240 pages) and an abbreviated booklet (60 pages). Alcorn offers this advice about grieving the death of a pet:
"If we regard pets as God-created companions entrusted to our care, it's only right that we should experience grief at their loss. Who made these endearing qualities in animals? God. Who made us to be touched by them? God.... We love animals because God created us – and them – to love each other...
We all know stories of pets who've risked their live and died for their owners, because the animals' instinct to love and loyalty outweigh their instinct for self-preservation. It is noble for a person to lay down his or her life for others, so animals who do the same must also be noble. We needn't be embarrassed to grieve their loss or want to see them again. If we believe God is creator, that he loves us and them, that he intends to restore his creatures from the bondage they experienced because of our sin, then we have biblical grounds for not only wanting but also expecting that we may be with them again on the New Earth.
Let's not "correct" our children and grandchildren when they pray that they'll be able to see their pets again. Then answer to that prayer is up to God. But he loves to hear the prayers of His children, and there is scriptural reason to believe He may answer those prayers. Remember too that our children's instinctive grasp of Heaven--and what we should look forward to there--is sometimes better than ours."
Each and every animal, though fallen, is one of God's masterpieces that displays his glory in some way. We are responsible to take dominion over animals, but we also must learn to appreciate each and every one as the handiwork of our heavenly Father.