Heck Tate wants to move Tom Robinson because he is worried that someone will try to hurt him.
The town is pretty stirred up over Tom Robinson’s trial. To most white men, he is just another black man. They do not know him personally. They don’t go to church with him or know his family. His children don’t go to school with theirs. When he is accused of rape by Mayella Ewell, it does not matter that there is no evidence. All that matters is that she is white and he is black.
Heck Tate, the sheriff, is worried that a mob of men might try to attack Tom Robinson while he is awaiting trial. Scout and Jem overhear snippets of conversation by a “crowd of men” around Atticus.
"...movin' him to the county jail tomorrow," Mr. Tate was saying, "I don't look for any trouble, but I can't guarantee there won't be any...."
"Don't be foolish, Heck," Atticus said. "This is Maycomb."
"...said I was just uneasy." (ch 15)
Atticus argues that no one in Maycomb will take his client from him. He also thinks the men won't drink on Sunday. Heck Tate is still worried about the men from "Old Sarem."
It turns out that (a) Heck Tate was right and (B) Atticus can handle himself. The mob does come to try to get Tom, but Atticus stands his ground. Scout and Jem try to intervene, and the mob's leader Walter Cunningham feels ashamed of himself and leaves.