The collar of John Doe's coat is alternately up and down between shots when Ann is persuading him not to jump off the roof.
During the montage showing John Doe Clubs gaining in popularity, close-ups of a map are shown with flags being pinned up for every new club. At the end of the montage, the camera pulls back to show a map of the entire US. However, the final map has substantially fewer flags pinned to it than what was shown during the close-ups.
When the Governor sits, the ear piece of his glasses changes position between shots.
When John Doe tries to jump off the building, he first throws away his cigarette and prepares to climb over the ceiling. Then there is a cut showing the other people looking for him saying: "I wouldn't do that if I were you, John." In the next shot, he holds the cigarette stump in his hand again.
After "John Doe" intrudes on D. B. Norton's dinner party and tells him off, Norton calls his newspaper and orders a special edition which will reveal Doe as a fraud. Doe takes a cab from Norton's house directly to the convention hall. Within minutes of his arrival there, a horde of newsboys appear with copies of the newspaper. It would be impossible to print an extra edition in such a short period of time.
Correction:
There isn't a plot hole, because D.B. Norton isn't saying nor implying that the newspaper will be printed from them on. He stated, before Ann is running after John Doe's in the Rain, "that he was prepared for this" and this does imply that the papers were already prepared, printed before. D.B. was a very rich man, callous, evil man. He foresaw all the problems in his investments.