Audio/visual unsynchronized: As Adam watches, the digital LED clock in his hotel room changes from 3:59 to 4:00 am, and makes a faint click as it does. Clocks of this kind do not make such sounds.
Continuity: Adam suddenly has a different car about halfway through the film.
Errors made by characters (possibly deliberate errors by the filmmakers): Sam Cahall states that the new law allowing him to opt for lethal injection applied only to inmates convicted after 1984. Actually it was just the opposite: Those convicted after 1984 could only have lethal injection. Before 1984 convicts could choose between lethal injection and the gas chamber.
Errors made by characters (possibly deliberate errors by the filmmakers): When Adam is giving his argument, he states "Men rea - to do the deed". "Men rea" is the intent portion of a crime. "Actus reaus" is the act.
Revealing mistakes: In the scene in which Adam visits the Klan gathering on the night before the execution the same extra (an older man in a baseball cap) can be seen in adjacent shots both behind Adam and the gang of thugs on the opposite side of the room.