"As they were going along the road, a man said to him, 'I will follow you wherever you go.' And Jesus said to him, 'Foxes have holes, and birds of the air have nests; but the Son of man has nowhere to lay his head.' To another he said, 'Follow me.' But he said, 'Lord, let me first go and bury my father. But he said to him, 'leave the dead to bury their own dead; but as for you, go and proclaim the kingdom of God" (Luke 9:57-60 RSV).
Taken out of context, the passage you requested—"Leave the dead to bury their own dead"—seems harsh and unfeeling. As with almost all of Jesus' teachings, it calls us to go deeper within, which is why I've added the preceding verses.
According to Jewish burial customs at the time, the body was immediately sealed in a tomb; there was an immediate seven-day period of intense mourning, followed by a month of continued observance. After about a year, when the flesh had fallen away, the bones were placed in an ossuary and re-entombed. So the disciple asking leave to bury his father could have been anywhere in this process, since the full burial was not considered complete until the year had passed. According to this interpretation, Jesus was suggesting that the would-be disciple was using the extended process as an excuse to avoid making changes in his life.
I think there may be more to it than that. Jesus is speaking, at this point in the Gospel of Luke, to many people whom are experiencing him for the first time. In the first flush of excitement they are assuring Jesus that they are eager to follow him 'wherever [he] goes' (v. 19). Jesus is saying that the work of truly expressing his teachings can be very challenging, and it will require a new set of priorities that may be at odds with the established priorities (including religious traditions) of the world around them.
So Jesus is suggesting that those who still believe in the reality of death can deal with its dramas and effects. For those who have awakened to their true spiritual identity, their Oneness with the infinite Power of God, following Jesus—following where that spiritual awareness takes them—is much more important than the trappings and drama of death could ever be.