…the subject of reincarnation was not addressed at all.
The Bible refers in many places to a single resurrection and the Last Judgement. Is there any evidence to support the claim that the doctrine of reincarnation was removed from the Bible?

Many people claim that a certain council took the teaching of reincarnation out of the Bible. Although the majority cannot name the specific council, they most frequently refer to the Second Council of Constantinople (A.D. 553) and the Council of Nicaea (A.D. 325).
In reality, though, at that time there were already many manuscripts in existence spread throughout the Roman Empire, which makes it difficult to imagine how someone could have collected all the extant copies in order to have all of them “corrected”. Today there are many surviving manuscripts from the time before the councils, whose text accurately corresponds to the currently accepted text of the Bible. (See: “Selection of Ancient Manuscripts of the New Testament”)
…the subject of reincarnation was not addressed at all.
There are also thousands of quotations of the New Testament in the writings of the Church Fathers from before the fourth century, which corroborate the reliability of the text of the Bible.
The two councils’ course of events is well documented by historians, and the records show that the subject of reincarnation was not addressed at all. Although the teaching of pre-existence (the existence of the soul before conception) was condemned by the Second Council of Constantinople (AD 553) in response to the teaching of Origen[1], the teaching of successive reincarnations is a separate matter and was not dealt with by the council.
While many people think that Origen believed in reincarnation, he himself writes in his commentary on the Gospel of Matthew:
Some one might say, however, that Herod and some of those of the people held the false dogma of the transmigration[2] of souls into bodies, in consequence of which they thought that the former John had appeared again by a fresh birth, and had come from the dead into life as Jesus. (Book X,20)
…In this place it does not appear to me that by Elijah the soul is spoken of, lest I should fall into the dogma of transmigration, which is foreign to the church of God, and not handed down by the Apostles, nor anywhere set forth in the Scriptures… (Book XIII,1)
Several church fathers and early Christian authors confirm this opinion (e.g. Irenaeus: Against Heresies Book II,33:1–2, Tertullian: Apology 48, Gregory of Nyssa: The Making of Man 28,3, etc.).
If reincarnation was really taken out of the Bible, it is not enough to assume that such passages were deleted from the Bible. The New Testament is interwoven with passages which speak about the fundamental Christian belief in Jesus’ resurrection and our own resurrection. All these passages referring to the teaching of resurrection (e.g. John 5:28–29), the last judgment (e.g. 2 Corinthians 5:10; Acts 24:15), eternal life and eternal condemnation (e.g. Matthew 25:46) would have to have been inserted, because the teaching of the last judgment clearly contradicts the idea of repeated reincarnations, through which people get a second, third (and so on) chance.
One passage in particular (Hebrews 9:27) clearly excludes the doctrine of reincarnation:
And just as it is appointed for man to die once, and after that comes judgement….
The oldest extant manuscript containing this clear statement is Papyrus 46, a document written in the second century AD, long before the Councils of Nicaea and Constantinople. Since the Enlightenment the Bible has ceased to be the exclusive privilege of the clergy. Today’s critical Bible scholarship (e.g. archeology, textual studies) would not be at all interested in keeping silent on reincarnation if there were even a trace of this teaching in the ancient manuscripts of the Bible. Additionally, there are surely many scholars who would relish the fame resulting from such a sensation.
The discovery of the Dead Sea Scrolls in 1947 raised the hopes of many that evidence for the belief in reincarnation would finally be unearthed. However, the biblical manuscripts found there have been closely examined and published in English. They are now publicly available to everyone who wants to read them. They comprise the writings of a Jewish community, whose concept of the after-life corresponds to the teaching of the Old Testament that physical death is final and is the end of life on earth (e.g. 2 Samuel 12:23; 1 Kings 2:1–2; Genesis 25:8) and that the righteous will rise and be with God in eternity (e.g. Daniel 12:2–3).
Finally, we want to encourage everyone who is searching for the truth and is in any doubt regarding the authenticity of the Bible, to start reading it for yourself with an open heart. We believe and have also personally experienced that, if we are open to listen to them, the words of Jesus can touch us very deeply and can shape our life and our way of thinking.
Origen of Alexandria (c. 185 – c. 253) was a Christian theological writer, biblical exegete, apologist, teacher and ascetic who wrote in Greek. (Wikipedia: Origen)
Transmigration: when the soul or spirit, after the death of the body, comes back to Earth in a newborn body—a concept identical to reincarnation.
Do not marvel at this, for an hour is coming when all who are in the tombs will hear his voice and come out, those who have done good to the resurrection of life, and those who have done evil to the resurrection of judgement. (John 5:28–29)
For we must all appear before the judgement seat of Christ, so that each one may receive what is due for what he has done in the body, whether good or evil. (2 Corinthians 5:10)
having a hope in God, which these men themselves accept, that there will be a resurrection of both the just and the unjust. (Acts 24:15)
And these will go away into eternal punishment, but the righteous into eternal life.” (Matthew 25:46)
But now he is dead. Why should I fast? Can I bring him back again? I shall go to him, but he will not return to me.” (2 Samuel 12:23)
When David's time to die drew near, he commanded Solomon his son, saying, “I am about to go the way of all the earth. Be strong, and show yourself a man, (1 Kings 2:1–2)
Abraham breathed his last and died in a good old age, an old man and full of years, and was gathered to his people. (Genesis 25:8)
And many of those who sleep in the dust of the earth shall awake, some to everlasting life, and some to shame and everlasting contempt. And those who are wise shall shine like the brightness of the sky above; and those who turn many to righteousness, like the stars for ever and ever. (Daniel 12:2–3)