The answers so far have been ... misleading at best. For some reason, debates about the historical Jesus on the internet often come down to people arguing either that Jesus never existed in any way at all or that the gospels are historically accurate documents. Neither of these is true. Jesus most likely did exist, but the gospels are not historically accurate.

 

Anyway, I recently wrote a post about the historicity of the virgin birth narratives in Matthew and Luke. Basically, the narratives are most likely not historical. Three Magi (who only appear in Matthew) never actually came to visit the infant Jesus. They never actually gave gifts to him.

 

In Bart Ehrman's The New Testament: A Historical Introduction to the Early Christian Writings (which I recommend if you're interested in learning about the New Testament and the historical Jesus), he writes about the wise men specifically in the section about Matthew (page 120 in the fifth edition I have):

Ancient readers would have recognized the Magi as astrologers from the East (perhaps Assyria), who could read the course of current events from the movements of the stars. These wise men are pagans, of course, whose astral observation have led them to recognize that a spectacular event has transpired on earth, the birth of a child who will be king.

The text never explains why Assyrian scholars would be interested in the birth of a foreign king. Perhaps their worship of him indicates that they understand him to be far greater than a mere mortal, king or otherwise.

...

What the Magi evidently do not know is where the child is to be born. The star takes them to Jerusalem, the holy city of the Jews, the capital of Judea. ... The Jewish leaders known perfectly well where the messiah is to be born: Bethlehem of Judea. They can even quote the Scriptures in support ...

 

So the story of the wise men may have had multiple purposes. It shows that Jesus' power is recognized even by pagans. It also shows the superiority of Scripture to astrology (people in the ancient world really did believe in astrology). Unfortunately Ehrman doesn't mention the symbolism of the gifts.

I'm sorry that I can't give you a fuller answer about the gifts specifically or about later Christians traditions about what happened to the gifts. But I thought as a starting point it would be helpful to clarify some of the historicity issues with the gospel narratives.