How does Heck Tate show support for Atticus, Judge Taylor, and others?

As an officer of the law, it is Heck Tate's duty to respect those who represent the laws of the county he serves, such as Judge Taylor and Atticus Finch. Outside of his official position, Sheriff Tate also affords personal respect to Atticus, Tom Robinson, and Boo Radley.

• Judge Taylor

On the day that the trial begins, Scout observes that Sheriff Tate is not in his usual attire when he comes to the witness stand. Instead, he is neatly dressed in a business suit, out of respect for the judge and the court of law. He answers questions honestly and directly. At one point in the trial when Judge Taylor interjects, "He's answered the question three times, Atticus..." Sheriff Tate remains compliant with the court process.

In Chapter 10 when Sheriff Tate is called because there is a rabid dog in the Finch's neighborhood, Heck Tate defers to Atticus to shoot Tim Johnson, who staggers down the street, because the dog may soon turn into the Radley yard and there is only time for one shot. 

"Take him, Mr. Finch....
"Don't waste time, Heck," said Atticus. "Go on."
"Mr. Finch, this is a one-shot job."

In Chapter 30 Sheriff Tate comes to the Finch house after the children are attacked; Jem's arm is broken, and Bob Ewell lies dead. Shortly after he arrives, Sheriff Tate respectfully tells Atticus that Jem has not killed Bob Ewell. However, Atticus does not believe him:

"Heck, it's might kind of you and I know you're doing it from that good heart of yours, but don't start anything like that."
"....Like what?" he [Tate] said.
"...nobody's hushing this up. I don't live that way."
"Nobody's gonna hush anything up, Mr. Finch."

Sheriff Tate goes on to explain, and Atticus finally accepts that Bob Ewell has died as a result of falling on his own knife, especially after Sheriff Tate claims he can prove it and does.

Sheriff Tate tells Atticus respectfully that he has solved the murder--"Bob Ewell fell on his knife," and there is no need to draw attention to anyone.

• Tom Robinson

"....Lived in this town all my life...Know everything that's happened here since before I was born. 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...."

• Boo Radley

Heck Tate tells Atticus that he has never heard that it is against the law for a citizen to try to prevent a crime from being committed. This, he adds, is all that Boo Radley has done. But, if Boo's actions are made public, Boo's privacy will surely be violated as the ladies of the town will start bringing him food and be curious about what is in his house, etc. Tate adds,

"...draggin' him with his shy ways into the limelight--to me that's a sin."

Clearly, Sheriff Tate is a considerate and fair-minded man who believes in justice and respect for all decent people.