After speaking with Mrs. Dubose, Scout, Jem and Dill walk with Atticus towards the Finch home. When the camera angle shifts, Dill is no longer walking with them, but is all the way on the other side of the house walking towards them.
When Scout and her brother are going to the pageant, she's dressed as a ham. In the closeups you can see that the "ham" is actually made of fiberglass. Fiberglass was invented in 1938 while the film takes place in 1932.
In the very beginning of the film, the adult Scout's narrates, "Maycomb was a tired, old town, even in 1932 when I first knew it." Moments later, she goes on to say that "Maycomb County had recently been told that it had nothing to fear but fear itself" in clear reference to Franklin D. Roosevelt's first inaugural address. Roosevelt, however, did not deliver this speech until March 4, 1933.
When Mr. Gilmer calls his next witness, he calls the wrong one (Mayella) to come first, which would have made the entire case scene non-crucial as it was in the book. Gregory Peck picks up on this superbly and improves the line which makes Mr. Gilmer re-think and correct his actions. This was all a mistake by Mr. Gilmer in the movie.
During the courtroom scene, there is a closeup of Atticus seated at his table during the prosecuting attorney's questioning of Mr. Ewell. In this shot, the film image is reversed (Atticus' hair and the position of the spectators behind him reveal this). In this same shot, the light at the back of the courtroom is turned on; both before and after this shot, it is off.
When Bob Ewell is on the witness stand and Atticus asks him if he ran for a doctor, Atticus is hovering over Ewell and his visible shadow behind Ewell reflects this. When the camera shifts to Atticus, he is a good 10-15 feet away. When the camera returns to Ewell, Atticus' shadow is still there.
While Atticus gets his papers together in the courtroom after the verdict you see a water glass next to the pitcher on the judge's desk. In the next shot, as he walks out, there is no glass, just the pitcher.
When Atticus is sitting in front of the courthouse door to protect Tom Robinson, the bottles in the top crate in the corner change from fewer bottles before the mob arrives, to more when the mob has left.
When Mr. Gilmer gets up to cross examine Tom Robinson, he walks past the seated Atticus at the defense table. You can only see the arm of the person as he walks by the table. The person who walks by is wearing a short-sleeved black t-shirt (he is probably a production stand in). When Mr. Gilmer reaches the witness, he is wearing the same light colored suit he has worn throughout the entire trial, which covers his entire arm.
While Atticus is giving his closing arguments standing before the jury, the camera briefly switches to Scout sitting in the balcony and then quickly back to Atticus who is now leaning on rail in front of jury instead of standing stock straight.
When the jury comes back into the courtroom with their decision (immediately after Jem's comment that it was a good sign that it had taken the jury two hours to reach a decision) and we see the procession of jurors walk into the courtroom (from Jem's point of view), no one is sitting at the defense table, nor is Atticus seen anywhere to be seen in the courtroom. However, in the next scene, Atticus is sitting at the defense table with onlookers in the background.
When sitting on the porch with the Judge, Gregory Peck is shown sitting on the porch swing with his left arm on the arm-rest. However, during the solo shots of the judge it is obvious that no-one is sitting on the swing - no arm on the arm-rest and the swing's chains are loose.
Directly after the scene where Jem and Scout are attacked while walking home through the woods, as Scout runs after the figure carrying Jem home, the trees and scenery can be seen through Scout in a ghostly fashion as if they were not originally part of the scene and were added afterward.
The goof item below may give away important plot points.
After the attack on Jem and Scout, Boo carries him towards the house and eventually is discovered standing behind the door in Jem's room. However, in the scene before he is discovered the door to the room is almost closed completely and then opened by Calpurnia from inside the room. Boo would have to have been visible to all in the room at that time.