Mr. Peabody says several times that he firmly opposes visiting a different part of one's own lifetime. But while fleeing from Troy, he says that he plans to change things so that their whole erstwhile problem never happened, which contradicts this policy.
When Mr. Peabody is dropping Sherman to school for the first time, he reminds Sherman that "I before E, except after C". Although this is a popular saying, there are in fact more examples that break this *weird* rule rather than follow it (there are 923 words in English that contain "cie", for example).
At around one hour into the movie, showing Mr. Peabody's thoughts on how to rescue Penny Peterson, there is a spelling mistake. "'Caclulate' total rope length needed." Also, from one of the number entries, there is a missing apostrophe.
In the second "Einstein on the Beach" scene, Mrs. Peterson's cocktail has a green umbrella. But when Sherman and Penny call Mr. Peabody, Mrs. Peterson's glass has a pink umbrella in it. Then, in the shot over Sherman's shoulder, when Mr. Peabody jumps down from his chair, Mrs. Peterson's umbrella is green again.
Inside the Trojan horse, after a comment about the smell is made, Agamemnon raises his arm to show his hairy armpit. But a few seconds later, and for the rest of the movie, he is shown to have hairless armpits.
The sewer cover falls into the tunnel and knocks out Robespierre. Round sewer covers can't fall into the hole they cover, since the hole diameter is smaller than the cover diameter, and can't fit it - regardless of its position.
Leonardo da Vinci is shown painting Mona Lisa on canvas. Mona Lisa is a plank of poplar wood. (While other slight historical inaccuracies are clearly deliberate and played for laughs, this one appears to be unintentional.)
In the Paris sewer Mr. Peabody asks Sherman if he smells anything. Sherman says he smells methane, but methane is odorless to humans. It's not clear whether a dog could smell it but Sherman certainly wouldn't be able to smell it. Commercial methane is given a distinctive smell so that leaks are detectable by humans because the naturally occurring gas has no odor.
At the end of the movie, Einstein is seen as solving the Rubik's cube, only to realize that there is one piece out of place. This is impossible on a Cube that has not been disassembled, as the minimum number of pieces out of place on a cube is 2 pieces.
When manipulating the Rubick's cube, Einstein is left with only one wrong square - a center square. As the central structure of the cube is a rigid six-post element that never actually moves, the opposite square should also have been the wrong color.
Early in the film Sherman says that the Leaning Tower of Pisa didn't always lean (suggesting that he caused it to lean). In fact the tower of Pisa has always leaned ever since it was built, due to an uneven settling of the foundation. The lean got progressively worse until structural modifications were put in place to reinforce it around the year 2000. It is now declared safe for at least 200 years. However, Peabody cartoons often contain deliberate inaccuracies for the sake of a gag (in the original skit he explains that they're seeing history as it "should have been"), and this in particular is a subtle reference to the Galileo skit from the original cartoons.
The goof items below may give away important plot points.
When Mr. Peabody and Sherman are entering the future wormhole at the end of the movie, upon reaching terminal velocity, the windshield cracks. The windshield isn't cracked in any other scene, as it should be.