Batman. Jim Gordon's sociopathic serial killer son James Gordon Jr. believes that empathy is a weakness.
The Big Bad in the first Blacksad comic book is a notorious Corrupt Corporate Executive who murdered his mistress in cold blood and uses his vast wealth to place himself above the law. When Blacksad tracks him down to exact vengeance, the bad guy taunts him that he wouldn't dare to perform a Vigilante Execution because Blacksad is held down by his moral scruples. The villain prides himself on being so cold-blooded that he would do anything for power and his own gratification. Blacksad blows his brains out, noting afterwards that he wouldn't have been able to go through with it if for not for this display of sheer haughtiness.
Sonata in Turnabout Storm mentions feeling pity for others (in other words, compassion) as a weakness. There's a catch though: She doesn't see it a weakness of character, but as one in a pragmatic sense. Being a blackmailer became difficult when she began to realize the damage she was causing to others.
Batman Begins has Ducard trying to talk Bruce into executing a criminal during his martial arts training, arguing that Bruce's compassion is a weakness that his enemies will not share.
In The Dark Knight, once Gotham's top crime bosses call a truce to take down Batman, they discuss the fact that Batman does not kill and how it is a reason to not really fear him at all. Batman later proves the mafia leader, Sal Maroni, wrong by dropping him from a non-lethal, yet quite painful height.
In Batman & Robin, Mr. Freeze zaps Robin, forcing Batman into a Sadistic Choice: "Chase the villain or save the boy." He concludes, "Emotion makes you weak. That's why the day is mine. I'll kill you next time."
This is more distaste for one virtue, but in The Karate Kid, Sensei Kreese delivers a denigrating speech about mercy to his students.
Mercy is for the weak. We do not train to be merciful here. If a man faces you, he is the enemy. An enemy deserves no mercy.
Faora: You're weak, son of El. The fact that you possess a sense of morality and we do not gives us an evolutionary advantage. And if history has proven anything, it is that evolution always wins.
Superman II. During the battle in Metropolis Superman spends a lot of time saving people from falling debris. General Zod shares an observation with his fellow villain Ursa.
General Zod: This "Superman" is nothing of the kind. I've discovered his weakness. He cares. He actually cares for these people.
In Flash Gordon, as Flash is going to be executed, Dale is crying. Ming the Merciless and his daughter have this exchange:
Aura: Look! Water is leaking from her eyes!
Ming: It's what they call "tears". It's a sign of their weakness.
Various Sith in the Star Wars universe believe that the Jedi's adherence to their Code (which promotes self-control and peaceful conflict resolution) is what makes them weaker than the Sith themselves (who value freedom from restraints above all else).
In Downfall, Adolf Hitler gives a speech during one of the dinners in the bunker that he believes it is natural and right that the strong should kill off the weak, citing how apes bash in the skulls of smaller ones during fights. He calls compassion towards other people a "natural sin" and morality a human flaw.
In A Song of Ice and Fire, Petyr "Littlefinger" Baelish remarks that Eddard Stark's honorable nature is his greatest weakness even though he treats it as his armor. This comes back to bite Ned when Cersei Lannister and Littlefinger exploit that Good Cannot Comprehend Evil and capture him, leading to his public execution by the new king Joffrey.
This is a common belief that is passed down through the current Lannister family. Even the otherwise good-intentioned Tyrion considers honor to be a way to death.
It could even be argued that this is the central theme of the entire franchise.
Voldemort considers love, friendship and anything related to compassion a weakness.
Lionel Luthor in Smallville believes in this wholeheartedly, and he has been teaching this to his son Lex his whole life. Lex initially tries not to follow and to be genuinely good at first, but later on, everyone's constant rejection of his attempts at goodness leads him to gradually embrace his selfishness and exploitation of others.
In Lois and Clark, after Lex Luthor analyzes Superman through a series of tests of his strength and speed, he remarks to his servant that he has found the chink in his armor; the fact he has morals.
In The Big Bang Theory, once Sheldon Cooper realizes that his girlfriend Amy had been making him more compassionate and open-minded he goes and attempts to break up with her, realizing that she had been weakening the strength of his own values of stubborness and complete disregard for others' opinions.
Discussed in Sherlock, with Sherlock asking his brother is his lack of caring about others is a good or bad thing.
Sherlock: Look at them. They all care so much. Do you ever wonder if there's something wrong with us?
Mycroft: All lives end. All hearts are broken. Caring is not an advantage, Sherlock.
The Cyber-Planner in Doctor Who episode "Nightmare in Silver" scoffs at the Doctor for having emotions, and for his willingness to make sacrifices to protect the children in his care.
In The Flash (2014), while Leonard Snart/Captain Cold acquires a "cold generator" that allows him to hurt The Flash, he considers his real weakness to be his focus on saving people instead of beating the bad guy. Snart uses this to his advantage by derailing a train and forcing the Flash to save everyone, then taking him from behind while the Flash catches his breath.
Warhammer 40K and Warhammer: As the Chaos Space Marines say it best, "Sanity... is for the weak!" Given that one of their gods is a literal embodiment of excess in every way, shape and form, they actually gain favor with him (her? It's complicated.) by corrupting the virtuous and the faithful. The others aren't much better, like the Chaos god of disease, plague and love being very generous with his gifts of boils and pestilence, while the god of backstabbers, mutants and sorcerers is hope incarnate. And then there's Khorne, who was the god of martial valor before becoming an enraged berserker who doesn't care if his followers, their enemies or non-combatants shed blood in battle note He is, however, very particular that his followers only claim the skull of a Worthy Opponent.
Mercykillers from the Planescape campaign setting were zero tolerance on injustice. Their credo is that 'mercy is for the weak, and the merciful should be punished'- they preferred to exact brutal, final punishments for criminals, and viewed mercy as a weakness to be purged from the multiverse.
This is one of Black's traits in Magic: The Gathering. It rejects all virtues that won't help it reach it's own personal goals and ambitions; Black isn't necessary evil, but it doesn't care about morals.
In Camelot, Mordred sings a song called "The Seven Deadly Virtues", saying that courage, purity, honesty, humility, diligence, fidelity and charity are "ghastly little traps" fit only for those more foolish than he.
In Dynasty Warriors 7: Empires. You can make Liu Bei— who's canonical obsession with ruling with Benevolence runs through his other officers as their hat— Evil, so to speak. If you make his Fame "Evil", he will eventually get a cutscene where he regrets being so virtuous.
At some point in Girl Genius, Gilgamesh Wulfenbach complains that every time he tries to be reasonable or show compassion, other people see it as weakness and attack him, forcing him to beat them up instead (in the process of doing exactly that). Then it clicks for him that the same happens to his father.
In the My Little Pony: Friendship Is Magic episode "The Return of Harmony", Discord manages to emotionally break the mane six (right before hypnotizing all but Twilight) by explaining how the virtues represented by the elements of harmony are weaknesses. Specifically, honesty causes others to know things you don't want them to know, laughter hurts the feelings of the one being laughed at, kindness opens yourself up to being abused by others, generosity prevents you from getting the things that you want, loyalty means nothing when you can't be in two places at once and have to choose between two loyalties, and in general harmony is pointless as its own elements lead it to collapse on itself.
In the short Education for Death, a German boy's teacher is horrified when he expresses sympathy for a rabbit killed by a fox in a story. Instead he claims that the fox is to be admired and the rabbit hated because according to the Nazis, Might Makes Right. After being the class fool for a day, Hans becomes a fanatic who screams that he wishes the rabbit would die.
David Xanatos, from Gargoyles said that love was his only weakness, after having to ally himself with the heroes. Goliath is sick to hear this.
Xanatos: So now you know my weakness.
Goliath: Only you would regard love as a weakness.