There are no Bible verses on this specific subject. But there is still a lot to say about the difference between human beings and animals. Whereas people are created in God’s image (Genesis 1:26-27), animals are not. People are able to enter into a relationship with God, whereas animals can’t. Human beings are engaged in religious activities, whereas animals are not. The longing for God is uniquely human.
God seeks reconciliation on a personal, individual level.
Even though all of creation was affected when Adam and Eve sinned, it is only with human beings that God seeks reconciliation on a personal, individual level. God desires a personal relationship with men and women and He opened the way for that through his Son, Jesus Christ (Hebrews 10:19-22). Each person who puts his or her trust in Jesus will be saved and reconciled with God (1 John 4:15). There is no Bible verse indicating that animals also need to put their trust in Jesus. It is on a more general level that creation, including animals, were reconciled to God. When Jesus died on the cross, He ‘reconciled to (God) all things, whether on earth or in heaven, making peace by the blood of his cross.’(Colossians 1:20)
Animals will remain part of God’s creation
The Bible gives no reason to believe that when animals die, they go to heaven. When a dog dies, that is simply the end of it. However, there is a passage in Isaiah that talks about animals living peacefully together with each other and with mankind (Isaiah 11:6-9). This probably refers to the situation on the new earth (although some scholars believe this is just imagery, referring to political powers). So probably there will be animals on the new earth, but there is no reason to believe that these animals are resurrected animals. It will be human beings and not animals who will be judged on judgment day. Jesus promises that He will raise up to eternal life each person who ‘looks on Jesus and believes in Him’ (John 6:40). This passage refers to human beings only. Also in Revelation 20:11-15 where the last judgment is described, only people are mentioned.
Taking care of God’s creation
God commanded man to have dominion over the animal kingdom (Genesis 1:28). This is one more indication that God has created mankind in his image and not animals. It also means that we should always treat animals in a decent way. They are certainly not the equals of human beings so we should not treat them as if they have human rights; but we should certainly give them the food and shelter they need. Christians should never be cruel to animals and never neglect them. We should be good stewards of everything God has entrusted to our care. ‘Whoever is righteous has regard for the life of his beast, but the mercy of the wicked is cruel.’ (Proverbs 12:10)