"Another disciple said to him, 'Lord, first let me go and bury my father.' But Jesus told him, 'Follow me, and let the dead bury their own dead.'" NIV translation
At first glance, Jesus' answer to this man's request may seem harsh and might even be seen as to advocate the breaking of a commandment (honor your mother and father). The reality is quite different from that perception but, in order to understand it, we must understand the Jewish burial practices of that day. According to tradition, the family was to bury (put in an accessible tomb) the body of their dead as soon as possible (usually within twenty four hours of physical death). The family would withdraw from public life for a period of seven days for mourning which is called shiv'ah. During this period of time, the mourning would take precedence over the religious obligations of a man so that he would not have to go to the temple and offer sacrifices, etc.
At the end of those seven days, a new period of mourning would continue for the family for thirty days and this was called shloshim. During this time, the family would go back to work and resume their normal religious obligations but they would usually avoid social gatherings.
At the end of shloshim, the family would go back to their normal routine but they would still "remember the dead in their prayers".
The body would stay in the tomb decomposing for a period of one year and, if anyone else from the family died, they would be added to the tomb along with those in the ossuaries and those that were waiting to be placed in the bone boxes. After this period of mourning (eleven months after shloshim), the family would go back to the tomb and gather the bones to put them in a box (ossuary). This act of gathering the bones was known as the second burial.
This period of decomposition became associated with the penalty for sin and gave rise to the concept of purgatory. The flesh was thought to be corrupt and so, when the flesh had decomposed, the belief was that the deceased was ready to enter their final rest with God.
The man in this passage was not asking to go and perform the duties of shiv'ah or shloshim but was wanting to go to the second burial of his father. With these facts in mind, we can see that Jesus was not being harsh but was telling the man (and us as well) that He is the only way to enter final rest (heaven) with the Father. His complete authority extends to the fact that He is the ONLY way to salvation. This was also continuing the transition of His ministry from teaching about the physical to the spiritual matters.