The literal answer would have to refer to the bible. The book of Genesis tells us that we are unique in God's creation, being made in His image and likeness. Also, though we and the animals have the breath of life in us, we were made to have dominion over the animals. It is reasonable to assume that our futures will be different, being of different flesh and having had different roles on earth.
We read in Isaiah 11:6-9 that there will be animals walking the earth during the Millennial Kingdom.
"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. And the cow and the bear shall feed; their young ones shall lie down together: and the lion shall eat straw like the ox. And the sucking child shall play on the hole of the asp, and the weaned child shall put his hand on the cockatrice' den. They shall not hurt nor destroy in all my holy mountain: for the earth shall be full of the knowledge of the LORD, as the waters cover the sea."
Those verses do not give us any indication one way or another specifically whether any of those animals will be our resurrected pets or whether they will be all newly created animals, or whether they will be animals completely different from any we could conceive of. After all, our normal picture of angels never prepared us for the passages in Ezekiel 10:10-12 with the cherubim of four faces and wheel within wheels. And a word of caution. Remember, a redeemed earth is not a resurrected earth.
After the thousand year reign of Christ on earth concludes and He makes a New Heaven and a New Earth (Rev 21, 22, 2 Peter 3:13, Isaiah 66:22) there is no indication that animals will be present with us. I take comfort in the Genesis flood accounts that He took care to provide for animals of the earth. He didn't make new animals after the flood, and He didn't give Noah only the animals he would eat and sacrifice, but He provided for all the kinds of animals. But the truth is, we simply don't know.
We do know for a fact that we will feel all joy in the new heaven, and the former things will not be remembered nor come to mind. (Revelation 7:16-17). I take that to mean that if our pets are not accompanying us in the eternal state, not only the crushing guilt of our sin will no longer be remembered and not only also our lost family members and friends will not come to mind, but also we will not remember our pets. Everything necessary to our happiness will be in heaven, and that will mean they will either be present or their memory will be absent.
The abstract answer to the question as to what happens to our pets indicates a growing awareness of the real impacts of imminent rapture. Many Christians who ask me this have been Christians for a lifetime, or at least many years. We always hear about the rapture and know it as truth and adhere to it as a doctrine, but peeking into the nuts and bolts of the mechanics of it has largely been absent from our thoughts. This is because until now it has not been obvious that we are the generation spoken of that will experience it. Though the rapture has been imminent for thousands of years, the conditions that immediately follow our departure haven't been.
But those conditions are present now and growing more glaring by the minute. We are the generation, (Mt 24:34) and the veil slips further from our eyes as knowledge increases (Daniel 12:4). Christians are waking up to the fact that the rapture could indeed occur in our lifetime, or next year, or tomorrow. As we gaze around at our world and see the absence of the church, we start picturing what it will exactly look like.
I have two indoor cats. When I'm raptured, whether it occurs when I'm home or at work, it means they will starve. They will be thirsty and alone and lonely and confused. Maybe a tree will fall on the house in one of the terrible storms and disasters prophesied to come upon the earth, and maybe they will be injured or escape defenseless into the world, only to be eaten by a larger predator. Maybe they will not escape the house but in their confusion and grief, turn on each other. I don't know. But it will be bad. The future does not look good for them.
Nor does it look good for the chickens in the pens and the cows needing to be milked and the dogs tied to the pole or inside the car while the master is inside getting coffee. I try not to think about it. It is just one of the many tragedies that will occur during this time. After all, Jesus Himself said "For then shall be great tribulation, such as was not since the beginning of the world to this time, no, nor ever shall be. And except those days should be shortened, there should no flesh be saved: but for the elect's sake those days shall be shortened." (Matt. 24: 21-22). It will be a very bad time, for not only the people but for the animals who suffer under the same corrupted earth that we do. I simply try not to dwell on it.
I take encouragement from something Texas Baptist Pastor Don Fawcett said, "For the creation was subjected to futility, not willingly, but because of him who subjected it, , in hope, that the creation itself also will be set free from its slavery to corruption into the freedom of the glory of the children of God. For we know that the whole creation groans and suffers the pains of childbirth together until now.” Romans 8:20-21. It seems abundantly clear from this verse that 1) the entire creation was subjected to decay and corruption due to sin, 2) animals and pets are created beings, 3) animals and pets are subject to corruption as part of the fallen world, 4) Christ, in his atonement, undid the sin/death principle for all corrupted creation, 5) at the eschaton, all things that have been effected by sin and corruption unwillingly will be restored. So, my conclusion- animals or pets in heaven? Animals? No doubt. Personal pets? I don’t know but it is certainly within the scope of biblical theology." I would like to agree with this.
I know God is good. His thoughts are above our thoughts and His plans are always working together for the good and His glory. It comes down to trust. I trust His thoughts and I trust His ways. After all, He sent Jesus to us. He sent the Holy Spirit to us. He sent the bible to us. He knows whither a sparrow falls and He knows our emotions and needs. Maybe He will send Fluffy and Scrappy to us! Heaven is perfection, and so, it will be perfect. I trust that with all my soul.