What happened to the apostle John after the Biblical account of his life ends?

We do not know what happened to the apostle John after the biblical account of his life ended. One of the New Testament gospels now bears his name as author, although it was anonymous until later in the second century, in which case he not only preached the gospel but wrote an account of the mission of Jesus. However, as John Carroll says in The Existential Jesus, at page 228, most scholars assume that John did not write the gospel. Burton L. Mack says in Who Wrote the New Testament, page 215, that before what is now known as John's Gospel was attributed to John, it had already become popular in gnostic circles where it was said that Cerinthus, the founder of a gnostic school, had written it. One way or another, we can say that the apostle John did not write the Gospel or epistles that became attributed to him.

Mack says (ibid, page 197) at some point, the Revelation to John of Patmos was associated with the writings of the Johannine school solely because of the common name. This is two stages removed from the apostle John, first by the assumption of common authorship with the Gospel in spite of very different theology and style, second by the assumption that the apostle John wrote the fourth gospel. Once again, there is no reason to believe the apostle John wrote this book, and many scholars now refer to its author as 'John of Patmos' in order to avoid confusion.

Thus, we can not say that John, son of Zebedee, spent his latter years writing. We have no reliable information as to where he went to preach, if at all. There is no historical evidence that he travelled to Ephesus where he took care of Virgin Mary or that he lived there until the end of his life.

We know that Paul died and Peter died but we don't know that John died. In John 21:21 Jesus told John to tarry till I come and in Revelations 10 John was told that he must prophecy again before many peoples nations and tongues. So I take it to mean, that John will be sent back to preach before the coming of Christ in our times

The Gospels states that John took Mary to "His own house" but the Gospel also give us a clue that John and his older brother James were very young by the way they behave. So from this, we can assume that John did not have his own house or wife or kids or estate but took Mary to his parents Salome and Zebedee house. Salome was Mary's younger sister making John Mary's nephew and Jesus' cousin. Remember that years later at about 25 years old John was still in Jerusalem not Ephesus or somewhere else. John is regarded as the youngest of the apostles and the Gospels give us a clue of his young age when Jesus had him resting on his chest at the Last Supper. John was not arrested at Golgotha giving us a clue also that he was young and harmless for Jews. He was to young to be considered a threat.

Another big clue that John did not have his own house, or wife, or kids, or estate is when he and James followed Jesus and asked him where he stayed. Certainly, nobody with house, wife and kids will go stay with another person.

What happened to the apostle John after the Biblical account of his life ends?

Our Lord upon his his Cross said these words to the Apostle St. John:

26When Jesus therefore saw his mother, and the disciple standing by, whom he loved, he saith unto his mother, Woman, behold thy son! 27Then saith he to the disciple, Behold thy mother! And from that hour that disciple took her unto his own home. - John 19:26-27

According to Catholic Church the main tradition is that St. John took the Virgin Mary to Ephesus after the Crucifixion of Christ. Others do exist but this one now has the favour of modern popes.

The Christian writers of the second and third centuries testify to us as a tradition universally recognized and doubted by no one that the Apostle and Evangelist John lived in Asia Minor in the last decades of the first century and from Ephesus had guided the Churches of that province. In his "Dialogue with Tryphon" (Chapter 81) St. Justin Martyr refers to "John, one of the Apostles of Christ" as a witness who had lived "with us", that is, at Ephesus. St. Irenæus speaks in very many places of the Apostle John and his residence in Asia and expressly declares that he wrote his Gospel at Ephesus (Against Heresies III.1.1), and that he had lived there until the reign of Trajan (loc. cit., II, xxii, 5). With Eusebius (Church History III.13.1) and others we are obliged to place the Apostle's banishment to Patmos in the reign of the Emperor Domitian (81-96). Previous to this, according to Tertullian's testimony (De praescript., xxxvi), John had been thrown into a cauldron of boiling oil before the Porta Latina at Rome without suffering injury. After Domitian's death the Apostle returned to Ephesus during the reign of Trajan, and at Ephesus he died about A.D. 100 at a great age. Tradition reports many beautiful traits of the last years of his life: that he refused to remain under the same roof with Cerinthus (Irenaeus "Ad. haer.", III, iii, 4); his touching anxiety about a youth who had become a robber (Clemens Alex., "Quis dives salvetur", xiii); his constantly repeated words of exhortation at the end of his life, "Little children, love one another" (Jerome, "Comm. in ep. ad. Gal.", vi, 10). On the other hand the stories told in the apocryphal Acts of John, which appeared as early as the second century, are unhistorical invention. - Catholic Encyclopedia

Blessed Catherine Emmerich adds this to the end of the life of the Virgin Mary:

After Christ's Ascension Mary lived for three years on Mount Sion, for three years in Bethany, and for nine years in Ephesus, whither St. John took her soon after the Jews had set Lazarus and his sisters adrift upon the sea. - Regarding Mary’s Age

Further more the House of Mary which St. John had built for Mary was discovered in 1818 and modern popes have given this location many indulgences, thus implicitly favouring this tradition as being true.

One of Emmerich's accounts was a description of the house the Apostle John had built in Ephesus for Mary, the mother of Jesus, where she had lived to the end of her life. Emmerich provided a number of details about the location of the house, and the topography of the surrounding area:

Mary did not live in Ephesus itself, but in the country near it. ... Mary's dwelling was on a hill to the left of the road from Jerusalem, some three and half hours from Ephesus. This hill slopes steeply towards Ephesus; the city, as one approaches it from the south east seems to lie on rising ground.... Narrow paths lead southwards to a hill near the top of which is an uneven plateau, some half hour's journey.

Emmerich also described the details of the house: that it was built with rectangular stones, that the windows were high up near the flat roof and that it consisted of two parts with a hearth at the center of the house. She further described the location of the doors, the shape of the chimney, etc.[8] The book containing these descriptions was published in 1852 in Munich, Germany.

Discovery in Turkey

On October 18, 1881, relying on the descriptions in the book by Brentano based on his conversations with Emmerich, a French priest, the Abbé Julien Gouyet discovered a small stone building on a mountain overlooking the Aegean Sea and the ruins of ancient Ephesus in Turkey. He believed it was the house described by Emmerich and where the Virgin Mary had lived the final years of her life.

Abbé Gouyet's discovery was not taken seriously by most people, but ten years later, urged by Sister Marie de Mandat-Grancey, DC, two Lazarist missionaries, Father Poulin and Father Jung, from Smyrna rediscovered the building on July 29, 1891, using the same source for a guide. They learned that the four-walled, roofless ruin had been venerated for a long time by members of the mountain village of Şirince, 17 km distant, who were descended from the early Christians of Ephesus. The house is called Panaya Kapulu ("Doorway to the Virgin").[14] Every year pilgrims made a pilgrimage to the site on August 15, the date on which most of the Christian world celebrated Mary's Dormition/Assumption.

Sister Marie de Mandat-Grancey was named Foundress of Mary's House by the Catholic Church and was responsible for acquiring, restoring and preserving Mary's House and surrounding areas of the mountain from 1891 until her death in 1915. The discovery revived and strengthened a Christian tradition dating from the 12th century, 'the tradition of Ephesus', which has competed with the older 'Jerusalem tradition' about the place of the Blessed Virgin's dormition. Due to the actions of Pope Leo XIII in 1896 and Pope John XXIII in 1961, the Catholic Church first removed plenary indulgences from the Church of the Dormition in Jerusalem and then bestowed them for all time to pilgrims to Mary's House in Ephesus.

House of the Virgin Mary

After the death of the mother of Jesus, St. John was eventually banished to Patmos where he wrote the Apocalypse.

A messianic community existed at Ephesus before Paul's first labors there (cf. "the brethren"),[Acts 18:27] in addition to Priscilla and Aquila. The original community was under the leadership of Apollos (1 Corinthians 1:12). They were disciples of John the Baptist and were converted by Aquila and Priscilla.[49] According to tradition, after the Assumption of Mary, John went to Ephesus. Irenaeus writes of "the church of Ephesus, founded by Paul, with John continuing with them until the times of Trajan." From Ephesus he wrote the three epistles attributed to him. John was allegedly banished by the Roman authorities to the Greek island of Patmos, where, according to tradition, he wrote the Book of Revelation. According to Tertullian (in The Prescription of Heretics) John was banished (presumably to Patmos) after being plunged into boiling oil in Rome and suffering nothing from it. It is said that all in the audience of Colosseum were converted to Christianity upon witnessing this miracle. This event would have occurred in the late 1st century, during the reign of the Emperor Domitian, who was known for his persecution of Christians.

When John was aged, he trained Polycarp who later became Bishop of Smyrna. This was important because Polycarp was able to carry John's message to future generations. Polycarp taught Irenaeus, passing on to him stories about John. Similarly, Ignatius of Antioch was a student of John and later appointed by Saint Peter to be the Bishop of Antioch. In Against Heresies, Irenaeus relates how Polycarp told a story of

John, the disciple of the Lord, going to bathe at Ephesus, and perceiving Cerinthus within, rushed out of the bath-house without bathing, exclaiming, "Let us fly, lest even the bath-house fall down, because Cerinthus, the enemy of the truth, is within."

It is traditionally believed that John was the youngest of the apostles and survived them. He is said to have lived to an old age, dying at Ephesus sometime after AD 98.

An alternative account of John's death, ascribed by later Christian writers to the early second-century bishop Papias of Hierapolis, claims that he was slain by the Jews. Most Johannine scholars doubt the reliability of its ascription to Papias, but a minority, including B.W. Bacon, Martin Hengel and Henry Barclay Swete, maintain that these references to Papias are credible. Zahn argues that this reference is actually to John the Baptist. John's traditional tomb is thought to be located at Selçuk, a small town in the vicinity of Ephesus.

John is also associated with the pseudepigraphal apocryphal text of the Acts of John, which is traditionally viewed as written by John himself or his disciple, Leucius Charinus. It was widely circulated by the second century CE but deemed heretical at the Second Council of Nicaea (787 CE). Varying fragments survived in Greek and Latin within monastic libraries. It contains strong docetic themes, but is not considered in modern scholarship to be gnostic. - John the Apostle (Wikipedia)

St. John died around the year 100 AD.

As a side note, there are a couple of points one could add here.

First, there are no known bones or other relics of St. John. Catholics have an ancient tradition of digging up of canonized saints and placing their bones under altars. We know where St. Peter's bones are, and several other apostles, but not St. John.

Second, Our Lord did seem to say to the Apostles that there were some present who would not "taste death" before the second coming. (Matthew 16:28) It's not definitive but is definitely intriguing to wonder if John is still with us, living privately somewhere.