(at around 11 mins) When there is a lock-down in the casino, Gloria rams into the metal door and her impression appears on the smaller door. In the next scene when the black vehicle crashes in, her impression is on the bigger door.
(at around 8 mins) In the casino Skipper looks through the periscope and says "I say we let it ride" and pushes the periscope up. A few seconds later the camera looks up at the chimp carrying the second chimp; and the periscope is nowhere to be seen.
(at around 10 mins) In the casino, to turn the lights off Julian pulls the power switch downwards. When Mort takes the switch while Julian dances, he pushes down for the lights to come on and up for them to go off.
(at around 44 mins) When Vitaly jumps through the ring, he starts spinning clockwise. The next shot he stands still and the background turns anticlockwise, from this view it should turn clockwise.
(at around 12 mins) When the animals get inside the Luxury Assault Recreational Vehicle and they crash into the Red Ferrari, there is a man in black suit just froze in front of the black car. In the next shot, the man and the car disappears.
(at around 13 mins) When tracking down Alex, Marty, Gloria, and Melman, Dubois mentions they never should have left the forest and now they must deal with her. However, the forest is not a common habitat for any of the animals used in the film.
(at around 16 mins) During the epic chase sequence in Monte Carlo, to get their Luxury Urban Assault Recreation Vehicle to go much faster, the Penguins utilize a nuclear reactor built into the engine with uranium-238 powering it. Later, when they crash on the roof of the Hotel Ambassador, the L.U.A.R.V. falls and there should be a nuclear explosion as a result.
(at around 6 mins) When they popped out of the water searching for the penguins Gloria has goggles and snorkeling gear. It's for comedic effect, but she's a hippo. She can breathe underwater.
(at around 50 mins) During Dubois's singing scene, she enters the hospital room that contains four lights but shoots six times leaving one on to begin her song. This is because she shoots three lights, one shot to break the glass and one to break the bulb for each of the three lights.
(at around 1h 9 mins) When Alex and his friends told the circus animals that they were from the Central Park Zoo, they were very upset, including the Penguins. However, the Penguins were the ones who bought the circus, and they were with Alex and his friends the whole time. Because of this, they should've been guilty for lying too. Clearly, this is a joke for comedic effect.
(at around 6 mins) In Monte Carlo, when the animals emerge from the water, they are not wet.
(at around 16 mins) US-style amber diamond-shaped traffic signs are seen when the penguin's car crashes through the road-work-barrier in Monaco. As all European countries, Monaco uses triangular red-and-white warning signs.
(at around 23 mins) The steam locomotive of the circus train is a North American style not European as are the heavyweight passenger cars.
(at around 5 mins) Considering their extreme difficulties in traveling much shorter distances unaided in the past 2 films (a problem which provides most of the series' humor), it is surprising that no explanation is given for how the animals traveled from southern Africa to Europe fairly easily.