Tim Johnson, of course, is not a man but a dog, and Heck Tate wants him shot because he has rabies and is foaming at the mouth. Rabid dogs can be dangerous, because they act irrationally and have been known to attack without cause. Since Tim is weaving down a neigborhood street, he poses a danger to others, and everyone is ordered inside. Heck doesn't want to shoot him because he apparently is not a crack shot, like his friend, Atticus. So, Heck decides to turn his rifle over to Atticus, who he remembers as being the best shot in the county. Heck knows that in spite of Atticus' long absence from shooting a firearm (which he does not even own), "One-Shot" Finch is still the best man for the job.
Tate did not want to shoot the dog because of the influence the pressure it had on him. Racism spreads quickly in maycomb and Heck did not want to go near such a thing.
atticus is the deadest shot in maycomb county