Does the Bible make any allusions to life on other planets? 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.