The Adventures of the American Rabbit suffers from this big time. The Big Bad's henchmen are a biker gang called The Jackals, who are... jackals. Several times during the movie, characters mention that no one should assume all jackals are evil just because of the actions of a few bad apples. All well and good, except that there are no good jackals in the movie — everyone is a member of the biker gang and is working for the main villain.
The main message of Barbie as the Island Princess is that you should marry for love, not wealth or status. At the end of the movie the King and Queen finally accept that their son loves Ro, and give their blessing to the marriage despite Ro not being a princess. Seconds later it's revealed that Ro is the lost princess of another kingdom, who everyone presumed had drowned in a storm as a child. So the movie cancels out its own moral in the space of two minutes. That has to be a record.
Subverted with The Book Of Life. The kids decide to free the animals about to be butchered. Then it looks like Manolo will use his bullfighting skills with the big nasty hog...but he does not, because the truth is that Manolo does NOT want to be a bullfighter.
Care Bears: Share Bear Shines opens with Oopsy needing to be rescued because he went to a dangerous place all alone. The other bears, including Share, admonish him for this, but not too long after that, Share goes off on her own, without telling anyone, to help a baby star get to Glitter City, where she's never been before and only has a vague idea of how to get there. The fact that she did exactly what she told Oopsy not to do is never brought up, not even when the others find her.
Delgo is an anti-war story. Two civilizations learn that they shouldn't fight each other... and then they team up to fight the armies of trolls, goblins and other monsters. Moreover, the fact that the monsters are Always Chaotic Evil severely undermines the anti-bigotry message.
FernGully has an incredibly anvilicious environmental aesop. Too bad, then, that the bad guys polluting and destroying the rainforest are stopped by the fairies living in it. So the aesop becomes less "help the rainforest, it can't help itself" and more "don't worry about it, the fairies can take care of themselves". Another issue with the fairies is that they're in the movie just to add an element of human interest to the story. The first problem with this is that it's implying that actual rainforests, where such creatures do not exist, aren't worth the attention of conservation. What's worse, though, is that the fairies live in a society based upon human ideals, which doesn't gel with the film's intended aesop that Humans Are Cthulhu—though arguably that aesop deserved to be broken.
Spoofed in The Incredibles bonus features on the DVD. One feature had one of the superheroes who was a Friend to All Children and worked regularly to keep them safe and educated give a speech about how important it is to stay in school, since the superhero in question dropped out. However, he quickly realizes he is mangling the aesop with him saying things like "stay in school, or you'll end up like me," since he is famous and well-beloved and has superpowers. He does not quite know how to proceed once he figures out that this is not sending the correct message.
Polish cartoon adaptation of Tytus, Romek i A'tomek is all about how advertisement and commercialization are bad; however the movie is rife with product placement, to the point that the good characters use an actual KFC restaurant as their spaceship.
Meet the Robinsons is particularly Anvilicious about its Aesop: don't worry about making mistakes because you can always learn from them and fix them later. The movie contains two plot-stopping lectures and a musical number to hammer it in. So, when confronted with DOR-15, Lewis solves the problem by declaring he will never invent her, causing a Temporal Paradox and removing her from existence. A quick and easy way to end the movie, but at the cost of undermining its Aesop. Right from the beginning, DOR-15 was still fully-functional, if only disobedient. The movie's solution prevents a viable third option: Instead of writing DOR-15 off as a failed invention too early, Lewis could remind his future self to either correct DOR-15's behavior or outright build a better one, allowing him to dispatch DOR-15 while still having his Helping Hat invention. Lewis also never demonstrates that he learned his roommate had needs and would be more conscientious about it. Meanwhile, the two characters who DO follow the Aesop's advice don't exactly get rewarded for it: Wilbur scrambles around trying to fix his careless mistake but only ends up making things worse and is eventually punished by his mother when he admits to it, while the Bowler Hat Guy keeps trying new schemes when the old ones fail and is consistently chewed out for his incompetence by DOR-15 and everyone else around him.
The short version: The film's Aesop is about getting better through learning from your mistakes. While Lewis laments that he makes the same mistakes over and over again, he ends up solving his problems by denying his mistakes (and potentially repeating them), rather than identifying and improving on them. Conversely, when Wilbur and Bowler Hat Guy do try and learn from their own mistakes, they end up making things worse for themselves.
Disney Toon Studio's Planes is about Dusty Crophopper, who dreams of competing against his heroes in a race around the world. Unfortunately, being a crop duster, he's not built for speed and must compensate with his considerable skill, setting up a lesson about how hard work and talent aren't as important as dedicated practice and perseverance. And then the rest of the cast donates new, superior parts to him so he can compete in the finale, to the point that by the time he's upgraded and ready to fly, he's not actually a crop duster anymore. Guess you can't be a winner unless you're built to win after all.
The movie is framed as a morality tale about the importance of family, but the movie doesn't support this at all. The two kids never reconcile with their parents, nor do they have to learn that family is more important than they thought; they save the dinosaurs from Screweye, and... their parents come back. It's worth mentioning that the Professor, one of the Aesop's main proponents, leaves his brother Screweyes to die at the end of the movie.
The guy who brings them back from the past has a time machine and food that can increase the eater's intelligence (though it may only work on animals with sub-human intellect). He uses this so that kids can meet dinosaurs. This might not be so bad, but he then leaves the dinosaurs to just wander free in the city, potentially causing panic and devastation.
The Ralph Bakshi animated film Wizards takes place in a post-technology future, and spends the entire film building up the conflict between a good, druidic wizard who lives in harmony with nature and who draws his power from all living things, and an evil wizard who's reinventing mass production, firearms and munitions, and whose conquering armies are threatening to plunge the world back into the chaos of technological warfare. The contrast between their philosophies keeps building until, at the end, they're finally facing down one another. And then the good wizard... shoots and kills the evil wizard with a gun.
Word of God is that the Aesop was supposed to be about propaganda. This is more easily seen afterwards, but the average viewer, without being told such, may gravitate to the more apparently obvious and familiar message that "technology is bad, being close to nature is good". This is the point: the movie sets up the Aesop and intentionally breaks it because the movie is denouncing the idea of letting the words of others do your thinking for you. The only reason technology is bad and nature is good is because everyone says so. The good wizard at the end is the only character who doesn't buy into the hype and recognizes that what the villain is doing is harmful, but it's the villain that's the problem, and is perfectly fine to use technology to solve it.