No they do not. Animals when they die "perish" - go to oblivion. See Psalm 49:20. People do not go to heaven either, if that is any consolation. Apart from Christ, as the Psalmist says, people perish forever just like animals. The only hope for life after death for those "in Christ" is the resurrection at his return. This is the sole (pardon the pun!) prerogative of people. Animals, no matter how well behaved, are not included.