What does the Bible say about reincarnation?
That is an easy question. The Bible says nothing about reincarnation except to deny the reality of reincarnation. In fact, the Bible quite specifically, denies the reality of reincarnation. Hundreds of passages could be used to prove the biblical reality that after this life we die and then face judgment for the things done in this one life. For example in Hebrews 9:27 “For it is destined for man to die once, and after that to face judgment.”
Now, it is true that some Hindus and others have tried to claim that the original New Testament writings mentioned reincarnation, but that references to reincarnation were later removed by the early church. This claim is not only without even a shred of evidence, it is downright absurd. We have dozens of manuscripts from the New Testament from the second and third centuries, and entire New Testaments from the fourth century, as well as thousands of quotes from the New Testament from the same time period. No reponsible scholar would ever support the evidence-free claim that the early church removed references to reincarnation, never mind that there are an unlimited number of unquestioned New Testament passages which prove reincarnation to be an unchristian idea.
We can safely dismiss this idea without prejudice.