I was recently informed that the LDS Church teaches that the Apostle John never actually died and is, presumably, still on earth somewhere today.
What is the Biblical and/or other basis for this teaching.
Also, if he is still alive and on the earth today, is there any church teaching regarding where exactly he may be?
There's an interesting passage towards the end of the Gospel of John, after Jesus's resurrection:
21 Peter seeing him saith to Jesus, Lord, and what shall this man do?
22 Jesus saith unto him, If I will that he tarry till I come, what is that to thee? follow thou me.
23 Then went this saying abroad among the brethren, that that disciple should not die: yet Jesus said not unto him, He shall not die; but, If I will that he tarry till I come, what is that to thee?
Here, John seems to be saying "no comment" on a certain rumor that had gone around in the church, leaving it deliberately ambiguous. The Book of Mormon actually gives an authoritative answer to the question, and this is where the LDS belief comes from.
From 3 Nephi 28:
1 And it came to pass when Jesus had said these words, he spake unto his disciples, one by one, saying unto them: What is it that ye desire of me, after that I am gone to the Father?
2 And they all spake, save it were three, saying: We desire that after we have lived unto the age of man, that our ministry, wherein thou hast called us, may have an end, that we may speedily come unto thee in thy kingdom.
3 And he said unto them: Blessed are ye because ye desired this thing of me; therefore, after that ye are seventy and two years old ye shall come unto me in my kingdom; and with me ye shall find rest.
4 And when he had spoken unto them, he turned himself unto the three, and said unto them: What will ye that I should do unto you, when I am gone unto the Father?
5 And they sorrowed in their hearts, for they durst not speak unto him the thing which they desired.
6 And he said unto them: Behold, I know your thoughts, and ye have desired the thing which John, my beloved, who was with me in my ministry, before that I was lifted up by the Jews, desired of me.
7 Therefore, more blessed are ye, for ye shall never taste of death; but ye shall live to behold all the doings of the Father unto the children of men, even until all things shall be fulfilled according to the will of the Father, when I shall come in my glory with the powers of heaven.
8 And ye shall never endure the pains of death; but when I shall come in my glory ye shall be changed in the twinkling of an eye from mortality to immortality; and then shall ye be blessed in the kingdom of my Father.
Doctrine and Covenants Section 7
Revelation given to Joseph Smith the Prophet and Oliver Cowdery, at Harmony, Pennsylvania, April 1829, when they inquired through the Urim and Thummim as to whether John, the beloved disciple, tarried in the flesh or had died. The revelation is a translated version of the record made on parchment by John and hidden up by himself.
1 And the Lord said unto me: John, my beloved, what desirest thou? For if you shall ask what you will, it shall be granted unto you.
2 And I said unto him: Lord, give unto me power over death, that I may live and bring souls unto thee.
3 And the Lord said unto me: 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.
4 And for this cause the Lord said unto Peter: If I will that he tarry till I come, what is that to thee? For he desired of me that he might bring souls unto me, but thou desiredst that thou mightest speedily come unto me in my kingdom.
5 I say unto thee, Peter, this was a good desire; but my beloved has desired that he might do more, or a greater work yet among men than what he has before done.
6 Yea, he has undertaken a greater work; therefore I will make him as flaming fire and a ministering angel; he shall minister for those who shall be heirs of salvation who dwell on the earth.
7 And I will make thee to minister for him and for thy brother James; and unto you three I will give this power and the keys of this ministry until I come.
8 Verily I say unto you, ye shall both have according to your desires, for ye both joy in that which ye have desired.
As mentioned in previous answers, the appointment of John to remain on the earth to preach the Gospel is analogous to the three Nephite apostles Jesus chose to do the same while visiting America. In fact, He said to them:
...ye have desired the thing which John, my beloved, who was with me in my ministry, before that I was lifted up by the Jews, desired of me.
3 Nephi 28:6
While records concerning the whereabouts of John are scarce, records concerning the Three Nephites are plentiful, such as as the following:
25 Behold, I was about to write the names of those who were never to taste of death, but the Lord forbade; therefore I write them not, for they are hid from the world.
26 But behold, I have seen them, and they have ministered unto me.
27 And behold they will be among the Gentiles, and the Gentiles shall know them not.
28 They will also be among the Jews, and the Jews shall know them not.
3 Nephi 28:25–28
If one can accept that the Apostle John and the three unnamed Nephite apostles received the same calling, rights, protection, and privileges, then it can be assumed that similarly John is hid from the world. John will be among the Gentiles and they will know him not, and John will be among the Jews and they will know him not.
Consider studying 3 Nephi 28 for more information and prophecies about the Three Nephites.
The basis for this teaching is, as noted in another answer, is found in John 21:21-23, where Jesus, speaking to Peter, sidesteps a question regarding John's future death. More direct teaching is found in Doctrine and Covenants 7, where Jesus is quoted as saying:
1 And the Lord said unto me: John, my beloved, what desirest thou? For if you shall ask what you will, it shall be granted unto you. 2 And I said unto him: Lord, give unto me power over death, that I may live and bring souls unto thee. 3 And the Lord said unto me: 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.
The LDS Church's summary of this is as follows:
The latter-day scriptures clarify that John did not die but was allowed to remain on the earth as a ministering servant until the time of the Lord’s second coming. (Guide to the Scriptures)
So yes, John is on the earth. As to his location, however, we don't have much in the way of details. The student manual to the Doctrine and Covenants, however, reports two appearances. The first is that Joseph Smith reported that the Apostle John (along with Peter and James) appeared to him and Oliver Cowdery in 1829. LDS historians generally hold that John, unlike the others, was not resurrected for this appearance, but rather "translated," that is, physically changed by God from a mortal to immortal being. As McConkie writes:
At that appearance, Peter and James were resurrected beings; John was translated. (Mormon Doctrine, pg. 572)
A separate appearance was reported by Heber C. Kimball at the dedication of the Kirtland Temple in 1836. (Whitney, Life of Heber C. Kimball, pp. 104)
More generally, the manual continues, there is one teaching from Joseph Smith that provides insight into why John's location is such a mystery:
In a conference of the Church on 3 June 1831, the Prophet Joseph Smith taught concerning John’s ministry: “John the Revelator was then among the Ten Tribes of Israel who had been led away by Shalmaneser, king of Assyria, to prepare them for their return from their long dispersion” (History of the Church, 1:176).
This is a reference to the Ten Lost Tribes of Israel, which were dispersed following the Assyrian conquest, and whose location is unknown. There is "speculation" that they went to the north, but "The Lord has not seen fit to reveal their location, [...] and until he does so, it is useless to try to identify their present locality." (Old Testament Student Manual, D5)
Thus, from Smith's teaching, John is (or at least was) "among" this dispersion of Israel, but we aren't meant to know where that dispersion is.
