They don't. I can't imagine where you got that idea, but it is incorrect. Mormons believe he died almost 2000 years ago (as do Catholics and other christians...)

They don't, they do preach that Apostle John is the one still hoofing it around. http://www.bcmmin.org/apost.html

I am sure you are speaking of John the Beloved and not Paul. Just before Jesus was about to leave the Apostles for the last time, he ask what he could do for them. Most of them ask to come to him after they died, that is all but John. We find this recorded in D&C 7;1-8.."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." The Bible also speaks of this, but is not so detailed...Later

No, not Paul. . They believe that the Apostle John (aka "John the beloved") is still alive and wandering the planet. . They also believe that three Nephites are also alive and wandering the planet. . Of course this negates the whole concept of the COMPLETE Apostacy that required Joseph Smith restoring the only true Church since there are, in the LdS worldview, four non-Apostates on Earth - but hey, that's another topic for another day!