Revealing mistakes: When Hurley is making his mad rush for his plane he clearly and visibly runs past a doorway with the word "Kamehameha" visible at the top. Most of the airport footage is apparently shot at the convention center in Oahu. Roberts Hawaii buses are also very noticeable within shots in the background.
Errors made by characters (possibly deliberate errors by the filmmakers): Despite being referred to as "The French Woman" and having a French last name, Danielle Rousseau speaks French with a heavy foreign accent. However, it must be remembered that she never claims to be French. The survivors assume she is French because of her transmission. "Rousseau" could be her married name, in which case she could be of any nationality.
Anachronisms: During the scenes in the 1970s, all of the spectacle wearing characters have an anti-reflective coating on their lenses (the greenish hued reflection). While these coatings are very common today, in the 1970s they were generally reserved for optical equipment lenses.