Interestingly, this old expression originates from the New Testament.When a man wished to follow Jesus and become a disciple, he asked if he could first bury his father, who had just died. But, Jesus replied,
Let the dead bury their own dead, but you go and proclaim the kingdom of God [Luke 9: 59-60]
This response's meaning is that the man must not put earthly duties before spiritual ones. After the children are attacked by the reprobate Bob Ewell and Jem left arm is seriously injured, Atticus discusses the matter with Sheriff Tate, alluding to Jem's having to go to court and testify to what has occurred.
"Mr. Finch, do you think Jem killed Bob Ewell? Do you think that?"
In the ensuing argument between the two men, Heck Tate explains that Bob Ewell fell upon his own knife because he disbelieves what Atticus proposes, that Jem stabbed Ewell; furthermore, he sees no need to involve Jem in this matter of the death of the despicable Ewell. The honorable Atticus does not want Jem to lose respect for his father if he covers the truth for his son. But, Tate insists on giving his report of the incident. He tells Atticus that this is his town, and he knows everything that goes on.
"There's a black boy dead for no reason, and the man responsible for it's dead. Let the dead bury the dead this time, Mr. Finch. Let the dead bury the dead."
In other words, let Tom Robinson "bury" Bob Ewell as an act of poetic justice, and the incident will be taken care of; in this way, Boo Radley with his "shy ways" will not be exposed to the gossip and cruelties of the public. The town can "move on" from the repercussions of the trial as all the injustice connected to it will finally end. The dead will take care of the dead, and the living will go on living in their own ways. Otherwise, it would be like killing a mockingbird to bring Boo to court.
Atticus finally understands the wisdom of Sheriff Tate and agrees with him.