This relates to the overall theme of the work, that it is wrong to cause harm to those who simply try to do good. Heck knows that if they open up a full scale investigation into the events that occurred Boo Radley would be brought into the public’s eye.
The ladies of Maycomb would be knocking down his door to bring him treats to praise his actions and Boo, who lives a reclusive life, would be put through misery for his heroic act of helping the children by fending off Bob Ewell and bringing Jem to safety. Heck’s belief is that they should let the dead, in this case Bob Ewell, bury the dead.
In addition to the idea of protecting "the mockingbird," Boo Radley, Heck Tate, sherriff in Harper Lee's To Kill a Mockingbird, seems reluctant to subject frail and mentally delicate Boo to the rigors of another court case that could easily end in travesty as did that of Tom Robinson. At any rate, Boo's trial could be, as Scout remarked of Tom's "a carnival."
Like Tom Robinson's trial, the interrogation of Boo--not to mention his father--would become an event that would bring out the curious and the cruel. As Mr. Tate himself tells Atticus,
To my way of thinkin', Mr. Finch, taking the one man who's done you and this town a great service an' draggin' him with his shy ways into the limelight--to me, that's a sin. It's a sin and I'm not about to have it on my head. If it was any other man, it'd be different. But not this man, Mr. Finch.
Here is Mr. Heck Tate's statement is another motif of Harper Lee's Pulitzer Prize winning novel, "The greater good is always the most important."
I think he does this for two reasons:
1. He is protecting Boo Radley from lots of attention, whether it could have been good or bad. Either Radley would have been tried for manslaughter, or he would have been celebrated for killing the bad guy. Either way, Boo did what was right in protecting the children. But, Tate wanted to protect him from the attention of the community and respect his private ways.
2. He is protecting Jem. Atticus seems to believe for a little bit that Jem might have been responsible for Bob's death. Tate knows that Atticus is entirely moral and would be okay with seeing his son tried in court for murder not because he wants that, but because he is a man of law.
Heck Tate lies about what happened because he thinks that that is the best thing for all concerned. There are a couple reasons why he thinks that.
First, he wants to protect Jem. Even though Atticus thinks that Jem can take it, Heck Tate thinks that that would be a heck of a hard thing for a kid that age to deal with.
Second, I think he is trying to protect Boo Radley. After Boo has been cooped up in that house for so long, it seems like it would be really hard on him to be in the spotlight.
Added to this, Ewell just isn't worth putting any decent people to any trouble...