I was reading Wikipedia about exaltation, and found a reference to people who are physically immortal but who do not advance to heaven. Where do they live? Are they on the Earth?
If you're talking about what I think you're talking about (without seeing the citation I can't be sure)...
Update / preface: I must clarify use of the term "immortal" -- in this answer, I mean it to be living beyond the natural age of life, or in other words, living for very many extended years, and not tasting of death. Upon further research, I see that Jesus says that John, and for that matter, the three Nephites spoken of below, will be changed from mortality to immortality in the twinkling of an eye and will not taste of death, though that process is, in fact, the death of their technically mortal bodies. Immortality in that sense means the state of being resurrected.
There are, that I know of, four immortal physical beings which have not yet advanced to "heaven" (more specifically, to their exaltation) according to Mormon doctrine.
There's John the Beloved who is an immortal being, still walking the earth (See here in Doctrine and Covenants for where the Biblical uncertainty on the topic is remedied; cf. John 21:20-23).
Mormons also have record, from the Book of Mormon, of three Nephites who were granted the same blessing as John from Christ as He visited the Americas, to walk the earth as physical immortals and minister.
3 Nephi 28:7
John and the three Nephites will all receive their full exaltation after their mission is complete at return of Jesus.
(Note: the newly-linked-to article on Wikipedia in the question references others, but Enoch, for example, and his family and city were taken from the earth, and we know very little about Moses, Elijah, and Alma and their commissions.)
38 Therefore, that they might not taste of death there was a change wrought upon their bodies, that they might not suffer pain nor sorrow save it were for the sins of the world. 3 Nephi 28:38 http://www.lds.org/scriptures/bofm/3-ne/28?lang=eng
We're looking for long answers that provide some explanation and context. Don't just give a one-line answer; explain why your answer is right, ideally with citations. Answers that don't include explanations may be removed.
My personal opinion as a Mormon citing scripture:
Basically I'd say that there are several beings or prophets identified in scripture, Old testament, New testament and the Book of Mormon, Doctrine and Covenants (D&C), and Pearl of Great price mentioning them.
1) John "the beloved" (see New testament references above).
2) "3 nephites"
3) Possibly Alma the book of Mormon Prophet
4) Possibly Moses, See scriptures below.
1) and 2 were at their request from Jesus. This "tarrying on the earth" until He comes again allows them to minister or "preach the gospel" on the earth and thus bring souls unto Him.
3) indicates he was a righteous man and not much is known about his possible death or burial
4) possibly was transfigured or did not taste of death so that the "melchizedek" or Higher priesthood keys could be preserved on the earth until they could be restored in future generations.
Various of these prophets appeared to Joseph Smith, Oliver Cowdery and others both before and after the official establishment of the church on 6 Apr 1830. The three nephites are said to have helped Joseph Smith and others in their translation work such as: assisting in carrying the gold plates and/or protecting them during the time Joseph Smith was translating them, assisting in farm work for those away helping so their families could have sustenance to live.
Also, in the Kirtland, OH Temple many prophets appeared such as Moses, Elias, and Elijah to Joseph Smith and others.
Also, in the Book of mormon we read that the 3 Nephites may "pray" and appear to whomsoever they will (if it's the will of Father in Heaven).
So, the 3 nephites had some kind of "transformation" upon their bodies so they would not die, but they have not yet been "resurrected" or changed from mortality to immortality and received their "full glory". There are 3 degrees of glory compared to the Sun, moon, and stars (Celestial, Terrestrial, and Telestial, respectively). And we believe that at some point nearly everyone living or who has lived on earth will inherit one of these 3 degrees of "glory" in a kingdom. This is likely yet in the future for most of the 4 beings aforementioned.
Also note: 3 ne 38:7-9 mentions that though they "shall never endure the pains of death", they have not yet changed from mortality to immortality. And that they while they "shall not have pain while in the flesh, neither sorrow save it be for the sins of the world;..." So, even though they are still living they can still experience sorrow for the sins of others and are not yet "immortal" per se.
Alma Chapter 40 has a good explanation of death and resurrection and the space between death and the resurrection.
Alma 45:19 mentions that the Book of Mormon prophet Alma, "may" have been "taken up by the Spirit" and may or may not have been buried.
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.
1–3, John the Beloved will live until the Lord comes; 4–8, Peter, James, and John hold gospel keys.
https://www.lds.org/scriptures/gs/three-nephite-disciples.p3?lang=eng&letter=t Three Nephite Disciples See also Disciple; Nephites; Translated Beings
Three of Christ’s chosen Nephite disciples referred to in the Book of Mormon.
The Lord granted to these disciples the same blessing granted to John the Beloved—that they might stay on the earth to bring souls to Christ until the Lord comes again. They were translated so that they would feel no pain and would not die (3 Ne. 28).
A.D. 420 - they were still living according to this:
https://www.lds.org/scriptures/bofm/morm/8.10-11?lang=eng#9 10 And there are none that do know the true God save it be the disciples of Jesus, who did tarry in the land until the wickedness of the people was so great that the Lord would not suffer them to remain with the people; and whether they be upon the face of the land no man knoweth.
https://www.lds.org/scriptures/bofm/3-ne/28.13-17?lang=eng#12 17 And now, whether they were mortal or immortal, from the day of their transfiguration, I know not;
11 But behold, my father and I have seen them, and they have ministered unto us.
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.
9 And again, ye shall not have pain while ye shall dwell in the flesh, neither sorrow save it be for the sins of the world; and all this will I do because of the thing which ye have desired of me, for ye have desired that ye might bring the souls of men unto me, while the world shall stand.