One of the stranger beliefs of Mormonism is that, at his own request, John the Apostle was granted immortality in order to do the greater work of winning souls to Christ and being a minister for those who shall be heirs of salvation who dwell on the earth. Doctrines and Covenants even goes as far as to cite John 21:22. It is alleged that the reason Jesus said "if I will that he tarry till I come, what is it to thee?" is because He had granted John immortality.
But if we ignore the fact that the Mormon caricature of John, this "flaming fire and a ministering angel", failed to prevent the alleged great, 1700 year long apostasy, and even to this very day refuses to show his immortal face, we need to note that the very next verse, John 21:23, specifically refutes the Mormon view. See, the Apostles themselves took Jesus' words to mean John would never die, but John is very careful to point out that Jesus did not say that. In context, Peter had just been given a vague description of how he was going to die (John 21:18-19), and Peter inquired about John, but Jesus effectively told him to mind his own business.
If John had genuinely been granted immortality, it seems unlikely that verse 23 would have been included. Why would John effectively say "some Christians think I'm immortal, but that's not what Jesus said..." if Jesus had said "Verily, verily, I say unto thee, because thou desirest this thou shalt tarry until I come in my glory, and shalt prophesy before nations, kindreds, tongues and people." (D&C 7:3)? The obvious answer is that Jesus did not say what is attributed to Him in Doctrine and Covenants. John eventually died, and he was clearly expecting to.
But if the Mormons truly believe John is still alive, they should have no problem convincing him to come out publicly, performing the same miracles we know the other Apostles could in their day, and declaring the prophethood of Joseph Smith. If they can convince him to do that, I'll commit my very life to the Book of Mormon.