The book of life
Whether names can be blotted from the book of life, and what that says of perseverance.
Revelation 3:1-6
1“And to the angel of the assembly in Sardis write: “He who has the seven Spirits of God, and the seven stars says these things: “I know your works, that you have a reputation of being alive, but you are dead.
2Wake up, and keep the things that remain, which you were about to throw away, for I have found no works of yours perfected before my God.
3Remember therefore how you have received and heard. Keep it and repent. If therefore you won’t watch, I will come as a thief, and you won’t know what hour I will come upon you.
4Nevertheless you have a few names in Sardis that didn’t defile their garments. They will walk with me in white, for they are worthy.
5He who overcomes will be arrayed in white garments, and I will in no way blot his name out of the book of life, and I will confess his name before my Father, and before his angels.
6He who has an ear, let him hear what the Spirit says to the assemblies.
Exodus 32:32-33
32Yet now, if you will, forgive their sin—and if not, please blot me out of your book which you have written.”
33Yahweh said to Moses, “Whoever has sinned against me, I will blot him out of my book.
What each tradition says
Rabbinic / Midrashic
The Jewish Encyclopedia defines the book of life as God's muster-roll, in which all the worthy are recorded for life. It says God has such a book, and that being blotted out of it signifies death, citing Exodus 32:32–33.
The Jewish Encyclopedia (1905), The Jewish Encyclopedia (1905), art. "Book of Life" — Opening definition
Patristic
Augustine says the book of life is not for reminding God, as if anything could escape Him by forgetfulness. It symbolizes His predestination of those to whom eternal life shall be given; His infallible prescience is the book, in which they are written, known beforehand.
Augustine, The City of God, Book XX (Nicene and Post-Nicene Fathers, First Series, Vol. 2; trans. Marcus Dods) — Book XX, ch. 15 (on Revelation 20:15)
Catholic
Aquinas says that those ordained by divine predestination to possess eternal life are written in the book of life simply, since they are written there to have eternal life in reality. Such people are never blotted out from the book of life.
Thomas Aquinas, Summa Theologica (Fathers of the English Dominican Province trans., 2nd revised ed.) — I q.24 a.3 — Whether anyone may be blotted out of the book of life?
Orthodox
No sourced position found. Checked the Chrysostom corpus (NPNF1-10 Matthew homilies; he left no Revelation commentary and none exists in the public-domain English corpus), Cyril of Jerusalem's Catechetical Lectures (NPNF2-07 — Daniel 12:1's "book" is cited in passing, with no teaching on blotting), and ANF vol. 7 (Victorinus of Pettau's Latin commentary is fragmentary, and the Eastern Apostolic Constitutions use "write them in the book of life" only as a liturgical prayer, stating no position). No public-domain Eastern statement on whether names can be blotted from the book of life found.
Reformed / Protestant
Gill holds that those whose names are written in the book of life shall never be blotted out. They will always remain in the number of God's elect, can never become reprobates, and shall never perish.
John Gill, John Gill's Exposition of the Whole Bible — On Revelation 3:5
Evangelical
Wesley says that if any even of these make shipwreck of the faith, he will blot them out of his book, even though they were written therein from, that is, before, the foundation of the world. The note points to Revelation 17:8.
John Wesley, Wesley's Notes on the Bible (Notes on the Revelation of Jesus Christ) — Note on Revelation 13:8 (the book of life)