Revealing mistake: When Jason climbs onto the rail of the fire escape, it is not attached to the wall - intentionally, it has not broken - but then in another shot, two seconds later, it 'breaks'. It pulls out of a visible hole in the wall, although it was never actually attached and there was no hole before. (00:23:25)
Revealing mistake: There is no snow at all on any surface higher than the second floor of the consulate, except for the sill Bourne is climbing on, and the fire escape.
Revealing mistake: During the Paris car chase scene a van gets knocked over - if you look in the van there is no-one in it. (00:52:45)
Revealing mistake: Just as Jason is returning to the car from the train station after dumping the bag in a locker, spectators can be seen watching the filming, which was apparently done on the sly and without consent. Nonetheless, the onlookers did notice the film shoot and followed the actors' actions, who are not doing anything in the scene to draw attention to themselves. Watch the black man to the left of Jason in the background, he follows the scene and his head turns as he watches. Then look behind Marie right after that and there are 2 or 3 people watching as well, in the background. This is very visible and very distracting. (00:50:50)
Revealing mistake: In the night scene where the Professor is driving in the rain to assassinate Womboosi, 2 people are visible in the car when it is shown from the front. Later he states that he always works alone so there is no reason for the other person.
Revealing mistake: When Jason climbs down the broken fire escape, the Embassy Marines are close behind him. One of them kicks open the exit door, he is wearing a pair of Navy blue trousers with a red stripe down the leg, along with black leather shoes (like most of the other Marines). When the Marine kicks the door open, it cuts to him looking down. His pants are now dark green or brown, along with different shoes. This shot being almost the same when Jason walks through the door and looks down.
Revealing mistake: When Jason is escaping from the embassy building the railing comes away from the wall. The man hanging on the building by one hand when the red bag falls is obviously a stuntman because he has curly hair. (00:23:32)
Revealing mistake: As Jason goes into the train to hide the bag of money, extras can be seen looking directly at him and the camera.
Chosen answer: They are octopii hung up to dry in the sun. A common sight at Greek islands. If you search Google, there are pictures of them. They are humorously mentioned at http://www.thevisualrecord.com/dbphotopages/photopage.php?photo_id=80.