Heck Tate gets this expression from something Jesus says. He tells his disciples that they should not worry about earthly things. Even if their father has died, they should let the dead bury the dead and they themselves should worry about their souls. This is from Luke, Chapter 9:
He said to another man, "Follow me."
But the man replied, "Lord, first let me go and bury my father."
60Jesus said to him, "Let the dead bury their own dead, but you go and proclaim the kingdom of God."
As to why Sykes makes Scout stand up, he does not exactly say, but it appears that he wants Scout to stand up out of respect for her father. Sykes, of course, is black and he wants to honor Atticus for what Atticus has done in defending Tom Robinson.
Reverend Sykes, the minister of the First Purchase African Methodist-Episcopal Church, requested Scout to stand at the end of the trial to show her respect for Atticus' spirited defense of Tom Robinson. The rest of the black gallery had already stood as Atticus prepared to leave the courtroom, and Reverend Sykes wanted Scout to understand what a special friend of the black population of Maycomb her father had become. The next day, Atticus awoke to find gifts of prepared food at his door from his admirers.
At the end of Chapter 21, Judge Taylor reads the guilty verdict. Scout watches as her father quickly shuts his suitcase and walks towards the south exit. As Atticus is passing, Reverend Sykes tells Scout to stand up because her father is walking by them. Reverend Sykes makes Scout stand up out of respect for her father. Reverend Sykes and the rest of the African American community respect Atticus's valiant effort defending Tom Robinson in front of a prejudiced jury. Standing as he passes by them is one way that the community demonstrates their respect and appreciation for Atticus.
Heck Tate alludes to a story found in the Gospel of Matthew when Jesus invited a man to be his disciple. The man told Jesus that he could not follow him right away because he had to bury his father first. Jesus responded by saying, "Follow me, and let the dead bury their own dead" (Matthew 8:22). Jesus was essentially saying that the spiritually dead are those who choose to put selfish tasks above spiritual duties.