What did Mary and Joseph do with the gold, frankincense, and myrrh? And when they were come into the house, they saw the young child
  with Mary his mother, and fell down, and worshipped him: and when they
  had opened their treasures, they presented unto him gifts; gold, and
  frankincense, and myrrh. (Matt 2:11)


I feel that it is pretty safe to assume that they at least used some of the gold to travel to Egypt, and perhaps even lived off of it while they were there. But I don't suppose any Josephologists or Mariologists have made any discoveries that perhaps give us some clues as to what exactly Joseph and Mary may have done with the gifts from the Wise men?
From the Wikipedia entry on the Biblical Magi:


  There was a 15th-century golden case purportedly containing the Gift of the Magi housed in the Monastery of St. Paul of Mount Athos. It was donated to the monastery in the 15th century by Mara Branković, daughter of the King of Serbia Đurađ Branković, wife to the Ottoman Sultan Murat II and godmother to Mehmet II the Conqueror of Constantinople. They were apparently part of the relics of the Holy Palace of Constantinople and it is claimed they were displayed there since the 4th century. After the Athens earthquake of September 9, 1999 they were temporarily displayed in Athens in order to strengthen faith and raise money for earthquake victims. The relics were displayed in Russia and Belarus in Christmas of 2014, and thus left Greece for the first time since the 15th century.


The above paragraph quotes a Russian TASS article, which says in part:


  The Gifts the Wise Men of the East, one of the few relics connected with the earthly life of Jesus Christ that have been preserved to the present day - gold, frankincense and myrrh are kept at the sacristry of the Agiou Pavlou (St. Paul’s) monastery on Mount Athos in Greece. The gold is in square and triangular plates bearing the finest ornament and measuring 5 by 7 centimeters. The frankincense and myrrh are in dark olive-like balls numbering 70. The box brought from Greece has part of the original Gifts: three gold plates with a thin filigree ornament with beads from a mix of frankincense and myrrh attached to it on a silver thread. The legend has it that shortly before her death, Virgin Mary gave these gifts to two righteous women. Later on, these relics were brought to Byzantium, and after the Turkish conquest in 1453, they were taken to Mount Athos by a Serb nun Mary.


Whether any of this has any historical truth to it is obviously pure speculation.