My bottom line is that I don't know the answer to this question.
I am considering it because yesterday was the day to deal with animal experimentation in a class, and because of Ptolemy Tompkins' article in the February, 2005 Guideposts. He is dealing specifically with pets.
Tompkins answers the question with a yes. What is his Biblical basis?
1) The covenant established with Noah states that
Gen 9:9 And I, behold, I establish my covenant with you, and with your seed after you;
10 And with every living creature that [is] with you, of the fowl, of the cattle, and of every beast of the earth with you; from all that go out of the ark, to every beast of the earth.
11 And I will establish my covenant with you; neither shall all flesh be cut off any more by the waters of a flood; neither shall there any more be a flood to destroy the earth.
12 And God said, This [is] the token of the covenant which I make between me and you and every living creature that [is] with you, for perpetual generations: (KJV, this and all Bible quotes, as the KJV is public domain.)
Tompkins says that this suggests that the covenant is not only with humans, but with animals, and is eternal.
2) Luke 3:6 says "And all flesh shall see the salvation of God."
3) Mark 16:15 says "And he said unto them, Go ye into all the world, and preach the gospel to every creature."
4) The same Hebrew words are used for living things in Genesis 1:21 and 24, as in Genesis 2:7.
1:21 And God created great whales, and every living creature that moveth, which the waters brought forth abundantly, after their kind, and every winged fowl after his kind: and God saw that [it was] good.
1:24 And God said, Let the earth bring forth the living creature after his kind, cattle, and creeping thing, and beast of the earth after his kind: and it was so.
2:7 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.
5) Isaiah 11:6 says "The wolf also shall dwell with the lamb, and the leopard shall lie down with the kid; and the calf and the young lion and the fatling together; and a little child shall lead them."
Tompkins does not say that the evidence is compelling. I don't think it is.
Referring to his evidence:
1) The covenant is a promise that God would not destroy the earth with a flood. It doesn't speak directly of Heaven at all, even for humans.
2) Luke 3:6 may be a figure of speech.
3) There is no scriptural evidence that the New Testament church made any attempt to preach to non-humans. (Supposedly, St. Francis did.)
4) The Blue Letter Bible gives access to Hebrew. Checking this shows that Tompkins is correct--the same Hebrew words are used. However, it seems clear that we don't really understand what a soul is. Even though the same words are used for life, or soul, for animals and humans, the Bible does seem to teach that humans are unique. The same first chapter of Genesis says that humans are to have dominion over the other creatures. (1:26, 1:28) Thus, even though there are properties shared between humans and other animals, they aren't all shared.
5) Tompkins believes that Isaiah 11:6 is about heaven. It is certainly not about the earth as it now is, where leopards aren't companions, and there is all too little peace. But the literal reading is more like an earthly, than heavenly, paradise.
I mentioned this subject to a student. She suggested that there were horses of fire, pulling the chariot of fire that drew Elijah to heaven. (2 Kings 2:11) She also mentioned that there were horses in Revelation. There are, indeed, four horsemen in Revelation, and Christ Himself appears on a white horse in 19:11-21.
God could have, of course, prepared these animals specially, as He prepared the fish that swallowed Jonah, the vine that shaded Jonah, and the worm that killed the vine. This doesn't have to mean that there is a heavenly stable, but it doesn't rule it out, either. I hope I find out!
The same genius suggested that I read Ecclesiastes 3:18-22:
18 I said in mine heart concerning the estate of the sons of men, that God might manifest them, and that they might see that they themselves are beasts.
19 For that which befalleth the sons of men befalleth beasts; even one thing befalleth them: as the one dieth, so dieth the other; yea, they have all one breath; so that a man hath no preeminence above a beast: for all [is] vanity.
20 All go unto one place; all are of the dust, and all turn to dust again.
21 Who knoweth the spirit of man that goeth upward, and the spirit of the beast that goeth downward to the earth?
On the one hand, this passage says that humans and animals die alike. On the other, it says that our spirit goes upward, and the spirit of animals goes downward. That sounds like you could argue it both ways from this passage.
A different student said that he had heard that a father responded to his young daughter, who had asked him about whether pets go to heaven, by saying that if having a pet in heaven was necessary to her eternal happiness, the pet would be there. I think that's a good place to leave this subject.
Thanks to both students, and to ALL my students. Without them, I wouldn't have had a job.