Many people believe that a pet goes to Heaven upon its death. I was curious, what did the Early Church Fathers (e.g., Augustine, John of Damascus, Tertullian, etc.) say about this?
I've read a lot of the early Church Fathers and don't recall them addressing this issue. A good site for reading their writings (for free) is at: http://www.ccel.org/fathers.html
Generally, the early Fathers are focused more on answering pagan criticisms of the Christian faith, arguing against heretical ideas, and generally building up the faith of believers. Ignatius, for instance, wrote letters of exhortation to the congregations he was passing by while on his way to Rome for execution. Clement of Rome wrote to the church in Corinth, calling them to remember the words of Paul which he had written to them. Justin Martyr wrote two Apologies to the Roman emperors, defending the Christian faith. Irenaeus wrote "Against the Heresies" to counter heretical beliefs and to defend the "rule of faith" (i.e. the creedal belief taught by the Apostles as they founded congregations).
My sense, then, in reading the Fathers is that the issue of pets was not something they were concerned about. I'm not sure how they would have taught about this issue, either, since the Bible is largely silent on the topic. What the Bible does depict, however, is that God is concerned for His whole creation, and that Jesus Christ is redeeming and restoring all of his creation. This is seen also, for instance, in the fact that God had Noah preserve the animals on the ark.
Thus, in Revelation, after the resurrection we see the "new heavens and new earth" that has been fully restored to the perfection in which God originally created it. Things will be perfect, but different there, and we don't quite know what it will be like, other than that it will be "very good" (cf. Isaiah 65:17, Isaiah 66:22, 1 Peter 3:13, Revelation 21:1, Matthew 22:29-32).
To the underlying question of "will we see our pets again in heaven or at the resurrection," the only answer I have is that God doesn't tell us; I suspect that's why the early Church Fathers don't seem to have addressed the question either. However, we trust that all things will be made right by God at the resurrection and that He will complete everything according to His goodness.