The short answer is: No, we are not told that in Scripture.
First of all, I should say I believe the “book of life” is a symbol of a spiritual reality. In other words, I don’t believe there is a literal book, but some might believe that, and that’s okay. Revelation 3:5 never says God blots out any names, only that he will never blot out names of believers in the book of life. It’s the book of “life” after all, not the book of “life and death.” Daniel also speaks of a “book” with believers’ names (Daniel 12), as opposed to other “books” that record judgement (Daniel 7). Revelation 3:5 is an encouragement that salvation is sure for those who trust in Christ. It is though their names are written in permanent ink and never erased, and apparently written down before the world ever began (Revelation 17:8). Revelation 17:8 says unbelievers’ names were not written down in the book. The purpose of this verse seems to be a confirmation of God’s certain foreknowledge, that he is not caught off guard by those who will always only ever hate God; He is not surprised or mocked by them; He knew what they would do, and He never wrote down their names “before the foundation of the world in the book of life of the Lamb who was slain” because he doesn’t make mistakes that He then has to erase. So in summary 1) Revelation 3:5 is assurance for salvation by grace through faith in Christ, and 2) Revelation 17:8 is a reminder of God’s certain knowledge of evil. God is “declaring the end from the beginning and from ancient times things not yet done” (Isaiah 46:10). Since God is not bound by time the future has always been as certain to Him as though it were all in the past. This is why David can say, “in your book were written, every one of them, the days that were formed for me, when as yet there was none of them.” Psalm 139:16.
Some have argued that the book mentioned by God and Moses in Exodus 32:33 is the book of life. Exodus 32:33 says, “But the Lord said to Moses, “Whoever has sinned against me, I will blot out of my book.” This should probably be understood to be another metaphorical book, like the books of judgment mentioned in Daniel 7, used here by the Lord to make a point. Ancient city records contained lists of their members whose names were crossed out when they moved or died. Since no one is without sin (1 John 1:8), Exodus 32:33 indicates either that all names were blotted out (since all sin), or that if some names aren’t blotted out, it’s not because the people didn’t sin, but for some other reason (namely Christ). The point of Exodus 32, which would have been understood by the Israelites, is that God hates sin and deals severely with sin, so that those who sin, die from it, and are “cut off from among his people” (Exodus 31:14). Every one of the Israelites died physically eventually, but since not everyone died a spiritual death (some are children of the promise: Galatians 4:28, Romans 9:8), and since everyone sinned (1 John 1:8), we know God is merciful. Everyone should have died, but some lived. God is gracious. So Pslam 130:3 reads, “If you, O Lord, should mark iniquities, O Lord, who could stand?” God overlooked ignorance. Since we know from Exodus 32 that we deserve death, and are instead shown mercy, we too ought to turn from sin and turn to Christ. “In the past God overlooked such ignorance, but now he commands all people everywhere to repent.” (Acts 17:30)
I hope this explanation helps. Although, I think we can all agree with Paul when he writes, “Oh, the depth of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways!“
 The second promise is that Christ will not erase the faithful saint’s name from the book of life. The book of life appears five other times in Revelation (13:8; 17:8; 20:12, 15; 21:27), and contains the names of believers written in it before the foundation of the world. This is in contrast to the “books” that record the sins of unbelievers, on the basis of which they will be judged (20:12–13). Note the “book” in Dan. 12:1 recording the names of the saved, and the “books” mentioned in Dan. 7:9–10 in the context of heavenly judgment. The promise I will not erase his name contains no inference that the names of the genuinely saved might for some reason be erased, but is rather an assurance that they will not. In both 13:8 and 17:8, the point is that the names have been in the book of life since the foundation of the world and thus cannot be erased, whereas those about to perish have not had their names thus written. Unbelievers are never associated with the book of life, but only with the books of judgment. Beale, G. K., & Campbell, D. H. (2015). Revelation: A Shorter Commentary (pp. 80–81). Grand Rapids, MI; Cambridge, U.K.: William B. Eerdmans Publishing Company.