What is the meaning of "let the dead bury their dead"? Matthew 8:21-23 says


  And another of his disciples said unto him, Lord, suffer me first to go and bury my father. But Jesus said unto him, "Follow me; and let the dead bury their dead."
  And when he was entered into a ship, his disciples followed him. (KJV)


What is the meaning of the dead burying their dead?
To me, this is a very offensive direction that Jesus gives.  And that isn't a bad thing -- frequently the Bible gives directions that are offensive to what we are accustomed to believing.  We should always take them seriously.

In this case, I think the message is that we must put following Christ above everything, even our most important worldly concerns.  In Hebrew culture, it is a son's duty to bury his father -- an important duty -- but Jesus is telling him to forget about that, to put following him first.  The options are "bury your father, and get around to following me in a bit" or "follow me".  As far as Jesus is concerned, there's only one choice.  It's quite similar to Matthew 19:29:


  And everyone who has left houses or brothers or sisters or father or mother or children or fields, for my name’s sake, will receive a hundredfold, and will inherit eternal life. 


As to the specific wording, I would suggest that "follow me" is contrasted to death.  So following Jesus is life, tarrying about with worldly duties is death.  So the dead in spirit are left to bury the dead in body: the living are following Jesus along the way.