ANSWER: Before we tackle the issue of pets and heaven, it is interesting to note that animals play a unique role in the Bible. After his creation, the first task graciously given to Adam was to name all the animals that were created (Genesis 2:19). The Bible records that not only did our loving Creator bring them to Adam, he was willing to refer to them by whatever name the first man gave them!
God certainly cared for the needs of the animals he created and intended some of them to be treated like pets. When he gave his laws to the children of Israel, he specifically stated that they, as well as humans, were to rest on the Sabbath day (Exodus 20:8 - 10). He also commanded that beasts whose labor helped man harvest food be allowed to eat some of the produce they worked on (Deuteronomy 25:4).
Regarding pets in heaven, we first need to consider whether humans will be there! When Jesus gave his well-known message called the Sermon on the Mount, he stated the righteous ("meek") would inherit the earth (Matthew 5:5).
Jesus is the only person so far who died and now lives in the heavenlies. Even after his resurrection, it was stated that King David had not yet ascended to his reward according to the message Peter gave Pentecost (Acts 2:29, 34).
Another issue is that nature is not an end in itself, unlike what many religious and philosophical environmentalists believe. Rather, the world of animals (even those we call pets) and plants serves the purposes of the master plan of God, who created nature for His overriding purposes.
One interesting thing to note is that when God creates a new heaven and earth there is no longer any need for seas (Revelation 21:1). Presumably, then, there will be no more whales or even fish. It may be that "saving the whales" from men driving them into extinction (not to say that is a good idea!) would only give them (say) a thousand or so years reprieve.
Animals will exist during the millennial reign of Jesus Christ. Those that are carnivores will be made docile like pets and eat plant-based food ("Wolves and sheep will live together in peace, and leopards will lie down with young goats" - Isaiah 11:6).
Animals, and any pets people have, will live on the earth and not in heaven, as only beings composed of spirit live there (see 1Corinthians 2:11). Men will live with them on the planet, but not in the eternal state created after God makes everything new again.