Why does Atticus take his jacket off during the trial?
This was a calculated move by Atticus to appeal to the members of the jury. He, and all of the other professionals in the room, have been in suit and tie throughout the trial. Despite the stifling summer heat, they have remained buttoned up and proper. By removing his coat and unbuttoning his shirt and vest, Atticus becomes just like any of the jurors sitting in that jury box.
"His voice had lost its aridity, its detachment, and he was talking to the jury as if they were folks on the post office corner" (Ch. 20).
His actions personalize him again. He is not, perhaps, Tom Robinson's attorney so much as he is now Atticus Finch, friend and neighbor. He is not talking above them and trying to teach them something; he is talking with them. He is engaging them at their level.
There is a risk in him behaving this way. The jury could view him as being too casual about the case and even unprofessional. However, his reputation is good enough that we know that is probably not going to be the way they view this, so it is a calculated risk worth taking.