On the morning after Henry's first trip to New Hampshire (which was made spontaneously, without any change of clothes or luggage), his shirt nonetheless changes repeatedly between shots.
When Susan tells Henry "none of us have run a presidential campaign before" she puts her left hand on his shoulder; in the next shot it's on his upper arm instead. (The discontinuity is interesting in light of the first lines of the film, which discuss the significance of where exactly Jack Stanton would place his hand on the arm of a person he was greeting.)
Near the beginning of the film, Stanton's tie is back to front, but in the next shot it's round the right way.
During the Mill speech you can see 2 TelePrompTers hidden in the crowd that John Travolta's character uses to give his dialog. In the reverse shot (back of his head) the TelePrompTers are removed from the shot.
Stanton's hand positions during speech to senior citizens.
When Libby and Henry are driving back from the Florida meeting, the reflections on their windows are in slow motion.
When Norman Asher suggests going on talk shows, his lips say, "Oprah," but the soundtrack says, "Montel."
During Thanksgiving at Jack's home, his mother asks everyone to sing "You Are My Sunshine" "in honor of all the southern governors" but her mouth says something completely different.
When Gov. Stanton is on "Geraldo" Richard Jemmons compares Henry to Libby "who actually goes crazy when her candidate turns out not to be the rock her church was built on", but his mouth does not sync with "the rock".
When Henry first meets Mrs. Stanton with the Governor at the airport in New Hampshire, the shadow of the camera is visible on all three actors as it pans around them.
When watching the New Hampshire returns, one TV station says that Governor Stanton is still out campaigning on the street until the polls close while the other simultaneously says 15 percent of the vote has already been counted. Votes aren't counted until after the polls close.