According to Mormonism, are there immortals on the Earth? 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


  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.


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.)