And it came to pass, that, as they went in the way, a certain man said unto him, Lord, I will follow thee whithersoever thou goest. And Jesus said unto him, Foxes have holes, and birds of the air have nests; but the Son of man hath not where to lay his head. And he said unto another, Follow me. But he said, Lord, suffer me first to go and bury my father. Jesus said unto him, Let the dead bury their dead: but go thou and preach the kingdom of God. And another also said, Lord, I will follow thee; but let me first go bid them farewell, which are at home at my house. And Jesus said unto him, No man, having put his hand to the plough, and looking back, is fit for the kingdom of God. (Luke 9:57-62 and Matthew 8)
There is controversy in the above verse that the man asking to bury his father was referring to waiting until his father died before following Jesus. Either way, whether the father was already dead and ready to be buried... or the man wanted to first go live with him until he died, is irrelevent. The next man wanted to bid his family farewell. Jesus is saying to let the (spirutally) dead bury their dead meaning those who are following Him (spiritually alive) are to do so and also not look back to be united with those who are not (spiritually alive). Jesus is showing the urgency of following Him today...now!
And Jesus, walking by the sea of Galilee, saw two brethren, Simon called Peter, and Andrew his brother, casting a net into the sea: for they were fishers. And he saith unto them, Follow me, and I will make you fishers of men. And they straightway left their nets, and followed him. And going on from thence, he saw other two brethren, James the son of Zebedee, and John his brother, in a ship with Zebedee their father, mending their nets; and he called them. And they immediately left the ship and their father, and followed him.(Matthew 4:18-22)
Again we see immediate urgency of leaving this old life to follow in a new life after our Savior. They 'immediately left the ship AND THEIR FATHER and followed Him'. This is letting the dead bury the dead, not hanging around waiting for the right time, getting things arranged at home, trying to bring people with us etc. Dropping our life today, to follow Christ for eternity.
See 2 Corinthians 6:2, today is the day.
The importance is repeated in Luke 9:23-26
And he said to them all, If any man will come after me, let him deny himself, and take up his cross daily, and follow me. For whosoever will save his life shall lose it: but whosoever will lose his life for my sake, the same shall save it. For what is a man advantaged, if he gain the whole world, and lose himself, or be cast away? For whosoever shall be ashamed of me and of my words, of him shall the Son of man be ashamed, when he shall come in his own glory, and in his Father's, and of the holy angels.
"Let the dead bury the dead" can also literally mean letting the spiritually dead bury their own (funerals) because what does light have to do with darkness? (2 Corinthians 6:14) If you study the funerals of today you will see man's traditions stemming from pagan roots, both which God warns us to avoid. (The pagan ritual of viewing dead bodies, lies that everyone goes to heaven, twisting God's word to suit their sermon, tons of money, glory to man etc...) Read the article on funerals here:
When you have a new life in Christ you are to leave all for Him which means not participating in the man made rituals that you grew up with, the ones that Jesus hates.
Beware lest any man spoil you through philosophy and vain deceit, after the tradition of men, after the rudiments of the world, and not after Christ. (Colossians 2:8)
If we are a new man in Christ (2 Corinthians 5:17) ALL things are passed away and ALL things become new, our life is new, not to be taken back to the old of who we were and what we did with those who are spiritually dead.
Notice these words in verse 60 (Luke 9)
Jesus said unto him, Let the dead bury their dead: but go thou and preach the kingdom of God.
"Go thou and preach the kingdom of God." That is what believers are to do, not participate in the pagan activities of funerals or going back to unite and live with earthly families who reject the things of God.
While he yet talked to the people, behold, his mother and his brethren stood without, desiring to speak with him. Then one said unto him, Behold, thy mother and thy brethren stand without, desiring to speak with thee. But he answered and said unto him that told him, Who is my mother? and who are my brethren? And he stretched forth his hand toward his disciples, and said, Behold my mother and my brethren! For whosoever shall do the will of my Father which is in heaven, the same is my brother, and sister, and mother. (Matthew 46:50)
When we are born from above (John 3:3) we now have a new family in Christ, those who are spirtually alive, therefore we are to let the dead bury the dead.
Note: this does not mean we are not loving to our earthly family. We are to witness the gospel to them, avoid those who claim to be of Christ but are living an unrepentant life, be kind and loving but the relationships will be changed when you become new in Christ. The closeness is gone, the conversations and activities have to be very carefully monitored and if you are truly leaving all to follow Christ and preach the gospel, you will have very little time for those who are not interested in the things of God and off you will go into your new life.