Revealing mistake: When the penguins emerge to carry the Penguin (character) to the water after he dies, the penguins are blatantly little people in costumes. Their "skin" crinckles unnaturally, and the spot where their arms meet their bodies is very flat and fake-looking.
Revealing mistake: When Batman is driving the Bat-boat, just after he dodges the two rockets the penguins launch at him, as the Bat-boat slams back down into the water in the background, the scale of the water-splash reveal that this effect was achieved using a miniature set and miniature Bat-boat model. The scale of the water splash, and the enormous "drops" that splash up give the effect away.
Revealing mistake: When the penguins launch the rockets/missiles in the end, watch carefully. In the second wideshot of the rockets launching (right after the bats begin to attack Danny DeVito), you can see two of the rockets hanging in midair, not moving. At the beginning of the shot, you can even see one of them spurting some fire. The rockets (attached to visible guided wires by the effects crew) got stuck in mid air right before the shot began, a few seconds before this second volley of rockets fire off.
Revealing mistake: When The Penguin (character) is first introduced, feeding the other (real) penguins, watch closely. While some of the penguins are real, it's blatantly obvious that some of them (particularly the ones to the left of the screen when Penguin says "Hi") are animatronic puppets, as they have very unnatural neck movements and don't waddle around like the rest. A few shots later, when Penguin says, "I believe the word you're looking for is...", you can even see an animatronic penguin on the left side of the screen that is unnaturally leaning slightly forward, yet never falls, revealing it's a puppet that was mounted to the ground.