What, if anything, does the Bible say either supporting the theory of life on other planets, or suggesting that there is none?
The Bible really doesn't have anything to say on the subject one way or another.
Historical writings suggest that some of early Christians believed that this world is only one of many that God has created, and particularly among Gnostic sects there were all sorts of interesting ideas on the subject, (for example, that the many worlds that God had created all had free interactions with each other, but not with us because this world was under quarantine, as no other people out of all God's creations were evil enough to murder their own Lord), but whatever the source of these ideas is, it's not found in the Bible.
The only verse I have ever heard being taken to mean this is John 10:16 - and that was humorously, not seriously.
As far as I can tell, the Bible is silent on the question of life that lives somewhere other than earth, with one exception -- angels.
Several passages in the Bible refer to the stars of heaven and the angels interchangeably, and other passages discuss angels in heaven or cast out of heaven to the earth.
My understanding is that such passages have led to a popular belief that every star (especially the "morning star" and similar stars later redefined as "planets") has an associated angel, much like Matthew 18:10 leads to the popular belief that every child has a guardian angel.
(This idea -- that each planet has at least one angel associated with it -- and other ideas that Mason Wheeler mentions -- is a big part of C. S. Lewis's fictional trilogy beginning with "Out of the Silent Planet").
The Bible as we have it doesn't give an absolute answer to the idea of life on other planets. I say "as we have it" simply because the books of the bible have changed throughout history.
Some argue that John 10:16 and first chapter of Ezekiel suggest life outside our planet and even possible visitation.
Others would argue that the omission of life being created on other planets in the creation story of Genesis proves we're the only planet with sentient life.
But why write about something that doesn't pertain to our personal walk with Christ? Telling us how others live for Him and worship Him on other planets would probably be something we couldn't comprehend. "For now we see through a glass darkly" (1 Cor 13:12)
But God wants to be worshiped (Deut 6:1-5). If this is the case, is it really far-fetched to think He would use the vast universe He created to hold a host of worlds that should worship Him?
Our limited minds have a hard time wrapping around the idea that there could be life on other planets that God also attends to. Many believe that since all the universe lives in sin (Rom 8:18-25) it suggests that Jesus would have to die repeatedly for each planet where sentient life exists. But the Bible states that "whosoever believes in him should not perish" (John 3:16) and that "by grace are ye saved through faith." (Eph 2:8)
Is it that hard to believe that God could decide to show another world His grace in a different way? Perhaps the sacrifice of life doesn't hold the same impact to them. I'm quite certain that God knows better than us what would get the message of His love and desire that none should perish across to others better than our limited minds would ever be able to fathom.
Or perhaps those other worlds still practice animal sacrifice as it was before the death of Christ. Perhaps those other worlds will experience a different thing to take them from animal sacrifice to salvation through faith and grace without that sacrifice.
Yes, the last part is just a guess ... an opinion. But it is not our place to say what method God can or cannot use to bring people to Him. It is foolish of us to limit Him. Ecclesiastes 8:17 states "Then I saw all that God has done. No one can comprehend what goes on under the sun. Despite all his efforts to search it out, man cannot discover its meaning. Even if a wise man claims he knows, he cannot really comprehend it."
If we can't comprehend what goes on here on our own planet, how can we possibly hope to comprehend what God is doing with the rest of His universe? I am sure the debate could go on indefinitely as to whether or not life exists on other planets. And while it's perhaps fun to debate such things, it definitely should not be our focus. "I press on toward the goal to win the prize for which God has called me heavenward in Christ Jesus" (Phil 3:14).
The Bible makes no obvious reference to aliens, and most people (including myself) think it makes no reference at all to aliens.
I think it is unlikely that God created life outside Earth for two reasons:
I think Pastor Doug makes a great case for life on other worlds: http://www.amazingfacts.org/news-and-features/news/id/528/bible-answers-with-pastor-doug.aspx
He highlights the idea that perhaps were the one fallen planet of many: “What man of you, having an hundred sheep, if he loses one of them, doth not leave the ninety and nine in the wilderness, and go after that which is lost, until he find it?” (Luke 15:4).
I wanted to respond to Wikis comments as well: --------"I think it is unlikely that God created life outside Earth for two reasons: - Scientific: because of the Fermi paradox which says essentially that if there is extra terrestrial life we should have seen evidence of it by now. - Theological: because Jesus died once only (Hebrews 10:10). So if there are aliens, would they need redemption? Apparently the answer is "Yes", if they have freewill. However, since Jesus has already died here on Earth, He couldn't die again for them. Therefore aliens with freewill cannot exist (though animal life might)."-------------
So to imagine just our world, just our few billion people, seems to diminish God's power in the universe and just increase my sense of self worth, when I would rather imagine a God who is unlimited in his creation.
And finally, am I the only Christian who would grin stupidly if the first friendly aliens we met turned out to have red skin, horns and a tail? And were ever so holy! :-) I'd love God to turn our terrible preconceptions on their heads!