During the fountain sequence where Beth picks up the coins, she enters with a bottle of champagne in hand. After a drunken speech, she enters the fountain, looks up at the statue and motions that she has her eyes on it holding the bottle in her left hand. The bottle disappears when she turns around and places both hands on her hips. It reappears inexplicably on the edge of the fountain when she starts grabbing coins from the water.
Nick's character was famously struck by lightning while playing football for Syracuse University in a rain storm. Syracuse plays in a dome so he would not have been exposed to the weather, however he could have been playing an away game. It never said the game he was struck by lightning in was a home game at Syracuse.
Nick's apartment scene obviously a set. When Nick first takes Beth back to his apartment, his friend comes to his door with a six pack of beer. Twice Nick slams the door shut in his friend's face, and each time the door slams, the entire surrounding walls bend and shake.
In the initial aerial shots of Rome the cars are driving backwards.
During the cab ride scene when Beth just arrived, the footage can be obviously seen sped up as the pedestrian on the left side of the screen is seen walking 3x then slowed down to 2x as the cab passed by.
In the Blackout restaurant, Nick grabs his chair and wine glass very easily as if the lights were on and he can see what he is doing.
When in the Blackout Restaurant while some people are using night vision goggles, shadows on the floor and reflections In the tables can be seen everywhere. Obviously filmed with a green filter.
When Beth eventually returns the coins to her stalkers she seems to know exactly which coin belongs to whom. However, this was never established throughout the movie. The only "coin" of whose owner she can be reasonably sure is Nick's poker chip (and even there she gets it wrong, because it was actually thrown into the fountain by Father Dino).
There is a theory, running through the film, that if somebody takes a coin out of the fountain the person who threw it in would fall in love with them. If this were true there would be just as many people taking coins out as throwing them in.