Plot hole: In the beginning of the movie, year 1865, Thomas Gates is asked to decode the playfair cipher for the K.G.C. This is later found and decoded by Ben Gates, who discovers "Laboulaye Lady" to be the answer. In the scene when Ben and Riley go to France to see the Statue of Liberty, there is something written by Laboulaye, ending with the year 1876. As that writing was only added in 1876, Ben Gates couldn't have referenced it in 1865.
Plot hole: When the characters are trying to escape Cibola through the central drain, the drain is actually, as Ben and Patrick imply, really just a room which dead ends, and they have to open the door to keep it from filling up. If this were the case, the entire city of Cibola and the room above (where Ben first says that all the water goes out somewhere) would have been completely filled up with water over the centuries. But in the movie the water had apparently been draining through that central "drain."
Plot hole: The idea that Ben Gates had to run away despite the President's being okay with the kidnapping does not make sense. The President could simply pardon him if necessary. However, when he ran away, he would likely be committing other crimes (fleeing and eluding police, reckless driving, etc.) that would not be federal crimes and so would not be pardonable.