The white University of Illinios football in Senator Irvings office (in the background by Janine Roth's head for much of the interview), changes positions during the interview. At times it's level on the shelf and at other times it's sitting crooked.
When Janine and Senator Irving are leaving his office he puts on his blazer and his collar flips up, he attempts to flip it down as he lowers his arms but fails. When the camera switches angles his arms are down and magically so is his collar.
When Senator Irving returns to his office after receiving a phone call, there is a lamp positioned slightly in front of the framed article (about him) that was written by Janine. When Janine was looking at the article previously, there was no such obstruction.
When Janine is riding past the White House in the taxi near the end of the movie, her view of the White House, as represented during her passing, cannot be obtained from a taxi since those streets are closed to vehicular traffic. Also, her view of the rows of graves in Arlington Cemetery during the same sequence cannot be obtained from a taxi since the paved paths in Arlington are closed to ordinary vehicular traffic as a taxi would be.
When Rodriguez and Arian are giving their presentation, they place letters of induction on the projector to show the class they enlisted. A letter of induction is a draft notice. The draft was over for over thirty years when the movie takes place, and since they volunteered, they would have used DD Form 4/1 "Enlistment and Reenlistment Document"
The movie shows the events in all 3 places beginning at 10am Eastern Standard Time, 6:30pm in Afghanistan, and 7am in California. In fact, Afghanistan is 9.5 hours ahead of Eastern Standard Time. If it were Daylight Saving Time, then it would be 8.5 hours, but the movie took place during winter and the text on the screen said Standard Time not Daylight time.
The title of the movie relates to a quotation noted by Robert Redford's character. However both the title and the comment by Redford's character are incorrect. The quote probably originates back to the Crimean War but was popularized during WW1 when a German General named Erich Ludendorff in complimenting the bravery of the opposing British soldiers yet criticizing British military command said "They are lions led by donkeys."
It is obvious that many of the scenes between actors are filmed with one or the other actor not being present. For example Meryl Streep's hair is different when the shot is from behind her. Clearly it's a double.