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Rei and Asuka of Neon Genesis Evangelion both struggle with this problem, albeit from different ends. On the one hand Rei's upbringing left her unable to articulate her emotions properly, on the other, Asuka deliberately tries to hold her emotions in check lest they overwhelm her.
To a lesser extent, Misato and Dr. Akagi go through this, which slowly drives them apart throughout the series. Akagi is almost entirely reliant on the Magi computers (and Commander Ikari) to make her choices for her, which always pick the most logical approach, and doesn't lose her head when things go awry. Misato has been known to send EVAs out on nothing more then "intuition" and is very concerned with them when things go wrong in battle, especially when it comes to Shinji. Eventually it comes south for both as Misato has to start making hard choices that could mean putting her job above her personal feelings and Akagi has an emotional breakdown when she finds Commander Ikari doesn't really love her.
Ai no Kusabi displays this theme with the social structure of Tanagura, but mainly the dynamic between Iason and Riki. Iason is the top ranking Blondy, that starts out as stoic, calm and rarely ever expresses emotion much like all of the Elites. Riki is the dark haired, bottom ranking, gangster that is defiant, wild and revered for it amongst other people from the slums. Somehow, they are attracted to each other. As the story progresses, Riki becomes more subdued but reluctant to accept this change, while Iason becomes more emotive yet accepts how he's Not So Stoic anymore and it's consequences.
In Undertaker Riddle, The undertakers are stoic and cold towards almost everyone, since they have lived so much with a job that makes them deal with dead people and evil spirits; they consider emotions as a sin since they can make you take a reckless and foolish move. Hayato, as Faust's succesor and future king must learn to balance this two things in his life to avoid becoming a cold automaton or an overemotional fool.
The idea behind the emotional spectrum of Green Lantern, where Green, as the centre of the spectrum, is the balance and thus control of the emotions the other colours represent. The Guardians of the Universe believe exhibiting emotions to be a sign of weakness, and lack of self-control, although with the amount of times their self-importance as the ultimate stoics has come back to bite them, many Lanterns (and readers) can and do argue the point.
Equilibrium comes down on the side of emotions. In the dystopia, emotions are demonized and people are made to take medications not to feel them.
A certain analysis on most of Christopher Nolan films is that they seem to give subtle messages about how goals are hampered by emotions when stoicism could have fulfilled the goals without much conflict.
The Star Wars novels carry both sides of this debate.
The Jedi order as it came to be in the prequel trilogy had boiled down the light side philosophy of self-denial and detachment into the following mantra.
The original Jedi Code, from before the Order had its final schism and permanently split, shows a bit of both sides (and arguably is closer to the Jedi ideal than the later Code):
"Emotion, yet peace;"
"Ignorance, yet knowledge;"
"Passion, yet serenity;"
"Death, yet the Force."
Subverted when the Jedi's rigid adherence to the Code—namely their ban on love—contributes to Anakin's fall to the Dark Side. In fact, when one compares the decidedly saner, more realistic New Jedi Order that Luke Skywalker founded, it's tempting to say that being damn near wiped out was a necessary lesson for the Jedi.
God Emperor Leto later attempted to use "passion" as a way to wind up humanity; by forcing them into a peace they couldn't control, they'd get sick of it and rebound a thousand times harder. Essentially treating emotional reactions as a spring he could wind up.
Inverted in the Harry Potter series, where Harry is consistently brash and emotional while Voldemort seems generally more cold and logical rational. It is said outright at one point that Harry's emotions towards Voldemort are what prevent him from going to the Dark Side. At the same time, however, Voldemort is generally considered the more powerful between the two and it is his moments of high emotion that inevitably screw him over.
Harry has the one emotion that Voldemort does not: love. He has the capacity to care for others and this is what prevents him from going to the Dark Side. Love verus evil is a major theme. In fact, Harry's sacrifice in the final book is the climax. However, Harry is not irrational in the least. He's emotional, he's rational, he's human, even if he's a bit angsty at some points, but that's not a stretch considering how many people die due to Voldemort. Harry vs. Voldemort is not emotions vs. stoicism, but love vs. evil. Besides, Voldemort is prone to fits of rage and only appears calm. He's more like a volcano about to erupt. With hate.
Also worth noting: aside from Voldemort, the main driver of the plot is Albus Dumbledore, who rarely exhibits anything but logic (and who blames his few mistakes on getting too emotional). The other plot driver, of course, is Snape, whose one long-buried emotional attachment is solely responsible for his Heel-Face Turn. So ... the general theme seems to be that love is good, but only if it's not Dumbledore's?
It's obsessive love that is explicitly described as dangerous — examples include Dumbledore's, but also Merope Gaunt's.
This is the theme of Sense and Sensibility, starting with the title. Elinor represents "sense," which then meant what it does now - having a good head on your shoulders and not letting your feelings carry you away. Marianne represents "sensibility," which meant more of a strength of feeling or something akin to Romanticism. Austen's sympathies are clearly more with the former; Marianne's strong sensibility is tested throughout the novel, and she is eventually forced to learn to be more like her sister.
The Dunyain from R. Scott Bakker's Second Apocalypse books are major stoics, literally. They were heavily based on the ideas of Greek Stoicism and the concept of the Sage. This contrasts with the Cishaurim whose powers are derived from passion.
Played straight in The Chronicles of Thomas Covenant. The New Lords found themselves far less powerful as spellcasters than the Old Lords because the New Lords were determined to keep their raw emotions in check and not fall prey to misusing their powers out of despair.
Then there are the Haruchai who like to bottle up their natural passion in favor for extreme stoicism and tend to end up highly judgemental of anyone who doesn't live up to their nigh-impossible standards.
One of the main conflicts between Drizzt and Entreri, though not based around morals. Entreri claims the superiority of his fighting style because he doesn't let himself feel emotions, while Drizzt claims his strengthen him. After repeated inconclusivefights Entreri is so desperate to prove his superiority that he has a duel set up in which he becomes completely enraged, while Drizzt keeps control of himself. In the end Entreri loses, while Drizzt dismisses the idea that this has proven either of them the better fighter. He considers it more due to luck, and the fact that Entreri is aging while Drizzt is comparatively barely out of his teens.
K.J Parker's The Scavenger Trilogy. One of the roots of evil explored is unbridled emotion. Stoicism is more respected. By the end however, the sinister extreme of lacking all drives befalls several stoic characters.
The Huanni embrace their emotions to the full, expressing them without restraint. They have an off-shoot race, Falorians, which are stoic and controlled (and pride themselves on such, being former slaves cast out of Huanni culture when they ceased being useful).
In the Star Trek: Titan novel Synthesis, an Andorian character is playing cards with a Vulcan, a Choblik and a Selenean (all races known for stoic logic in contrast to Andorian passion). Inevitably, she gets very, very frustrated by their exaggerated analytical approach to the game.
In Warm Bodies, it is inverted where it is the protagonists' passion for life (whether zombie or human) is what gives them the power to change the world while the antagonists' stoicism dooms them.
Live Action TV
Angel: Angel and Wesley often lock horns over the best way to handle cases. Wesley always finds a way to outmaneuver him, with disastrous results. One of these guys is too clever by half.
Somewhat inverted in Bones: Zack Addy, a character with apparent Asperger's and second to only Brennan in the logic department, sides with the serial killer called the Gormogon due to his own logic, believing the killer's to be infallible. Meanwhile, highly emotional characters such as Booth and Angela remain on the good side.
Although, in the end, Bones' own logic trumps Zack's by arguing (logically) in favor of emotion and attachment to other people.
And after that Butt Monkey Sweets proves that Zack's feeling of guilt (he considers himself a killer because he was certain that he was willing to kill) is based on severely flawed logic.
Kendra: Emotions are weakness, Buffy. You shouldn't entertain them.
Buffy: Kendra, my emotions give me power, They're total assets!
Kendra: Maybe, for you. But I prefer to keep an even mind.
The two older Bass men on Gossip Girl keep telling Chuck that he is weak because he loves and therefore is ruled by his emotions. Only by being emotionless and stoic can he ever achieve greatness. Chuck of course buys into this, even telling his girlfriend "I can't let my feelings cost me all that I've built" after he sold her to his uncle in exchange for a hotel.
On NCIS, early Ziva aggressively shut downs her emotions, and the other characters call her on it. Highlighted in the two-parter episode "Hiatus", where Gibbs in seriously injured, and the other characters are furious that she doesn't show how upset she is.
The titular character of Sherlock at first steadfastly believes that emotionless and stoic is the best way to ensure success in his chosen field as a consulting detective, though he later begins to have doubts. His elder brother, however, is adamant.
Sherlock: They all care so much. Do you ever wonder whether there's something wrong with us? Mycroft: All lives end; all hearts are broken. Caring's not an advantage, Sherlock.
In the sci-fi show Stargate Atlantis co-commanders and friends Elizabeth Weir and John Sheppard can fall into this dynamic. When facing an obstacle, Elizabeth is logical, while John is driven by impulses. Double reversal of the trope as she is a female civilian: associated with Emotion. He is a male soldier: associated with the Stoicism.
A common theme in Sirens with stoic Stuart intent on conquering all his emotions and psychological defence mechanisms but to the annoyance of the local therapist.
The Vulcans were sophisticated and clever too - they clearly got as far as interstellar travel before destroying themselves. Some suspect that the proto-Romulans probably agreed with the followers of Surak that something had to be done about Vulcan emotion, but strongly disagreed over exactly what. Regardless, they have clearly found some way to manage their emotions, since they are not nearly as warlike as the ancient Vulcans appeared to be. This does somewhat belie the oft-repeated Vulcan assertion that logic and emotion cannot be reconciled, but then again, the Romulans are no longer entirely the same species - perhaps they engineered the more volatile emotions out of their population?
In the Starfleet Battles spin-off universe, Romulan Culture is stated as having a principle of "Unifying Duty" to serve as an outlet for aggression and passion, where individuals set aside petty jealousies and personal ambition in service to the greater good of the species. While this has allowed the Romulans to avoid descending into barbaric self-destruction, it has also meant they use other species as an outlet for their aggression.
Star Trek: Enterprise had an episode where the crew was surprised to meet a sect of Vulcans who believed that as long as you were careful not to let it get away from you, actually emoting was not bad in and of itself; they also ate meat, although I'm not precisely sure how that's totally relevant.
Vulcans have been demonstrated in an earlier episode to be culturally vegetarian. The fact that these Vulcans eat meat helps to demonstrates that they are unusual and do not subscribe to the same set of social tabboos as mainstream Vulcans.
Supernatural has several examples, most of them inverting the usual trope:
Sam spends half of the 6th season soulless, which basically means that he's cold, logical, rational and highly efficient as a hunter without any nasty emotions to hamper him on the job. Oh and he also shoots innocent hostages, tries to kill Bobby and lets his brother be turned into a vampire.
Castiel is the one to get most blatantly morally corrupted.
House refuses to meet his patients, because remaining impersonal allows for objectivity. He criticizes Cameron and Wilson for becoming emotionally involved with patients, since it clouds their medical judgement.
The subject of Rush's "Hemispheres", which tells of a civilization's struggle with balancing emotion and logic.
The greatest irony is that if there is any hope for humanity at all, said Messiah will ultimately end up literally becoming God (Belief is literally power. And there are countless trillions worshipping the Emperor.) More accurately, he will most certainly become the fifth Chaos god if allowed to die and will end up being warped to fit his peoples' perceptions of him rather than what he was when he was alive. Stressing secularism while demonstrating the vastest Psychic Powers in human history was rather contradictory anyway.
The craftworld Eldar are extreme stoics. Their passion once created the chaos god Slaanesh and in the process nearly wiped out their race. Now, they follow the "path," the practice of complete self-denial and total dedication to the perfection of a single skill.
And if they dare to be too happy, the Chaos God Slaanesh (who destroyed the original Eldar civilisation) rips their soul out instantly. Anyone else feel like practicing self-denial?
Red's conflict with it's other enemy, White, also plays with this, but with an interesting side effect. White is all about duty and order; at it's best, it's The Cape, but at worst is Knight Templar. Red's nature as the colour of passion means that it considers duty and order to be opressive, while White sees Red as chaotic and destructive. This generates a conflict of Duty versus Empathy, as White feels it has obligations to the greater good while Red cares about people that emotionally stimulate it (be a friend, a lover or that homeless orphan down the street).
The main characters of Killzone stand on both sides of this, with Templar and Rico be emotional and Luger and Hakha be stoic.
In Space Siege the inversion of this trope essentially has a three-way with the Karma Meter and Cybernetics Eat Your Soul. The setting doesn't have augmentations in the proper sense—instead, you cut off and replace entire limbs with more powerful robotic equivalents, in the process losing your ability to empathize with normal people, which is a Very Bad Thing. (Word of God compares the process to getting addicted to cocaine.)
In I Miss the Sunrise, people apparently repressed their emotions in the past, though they're resurfacing now. The characters tend to treat them as both a blessing and a curse, instead of planting themselves firmly on one side. Most characters do lean towards one or the other. The personality bars, especially the mentality one, also represent this.
The Multiple Endings seem to be connected to this. The optimist ending is sentimental, the pessimist ending is rational.
In The Order of the Stick, V's recent story arc exemplifies the dangers of both sides of this pretty well actually. She/he pulls away from the group and emotionality (what little he/she had to start) and dives head first into repression and stoicism after the battle of Azure City. Eventually this leads to some pretty distressing lack of empathy, isolation, and eventually splitting of on her/his own to pursue the power needed to save the world. But when his/her family is suddenly threatened because of V's own past actions, her/his emotions come storming back, overwhelming his/her rationality and leading to some very distressing situations, possible Moral Event Horizon, and flirtation with permanent alignment shifting (to EVIL).
Word of God has it that this was originally meant to be the point of The Wolf's Will; unfortunately they lose their stoic after the first chapter
And find a better stoic.
Used in Transformers: Beast Machines - the Maximals (and Megatron) must master their emotions in order to transform, and if they lose self-control, they are forced back into beast mode. On the other hand, Megatron advocates the abolition of all emotions and individuality from Cybertron so that can become a god.
Avatar: The Last Airbender: "The Beach." Zuko and Mai's fight is basically this. Zuko is all emotion and he lets his temper and impulses get away from him. Mai is all stoicism to the point where she doesn't seem to care about anything and can't express any emotion. Eventually they come to understand each other, with Zuko learning to think more calmly and rationally and Mai learning to be more free with her feelings.