Red Garden deconstructs this, where the pressures of fighting monsters in the middle of the night cause problems for the main characters' social lives.
Though Digimon Adventure avoided this one, Digimon Adventure 02 waltzed right into it. A team of 11-year-old students who go to school during the day and face off against monsters and the evil emperor of another world in their spare time? Textbook.
It has one interesting subversion: said evil emperor is another eleven-year old student. But since he's evil he's allowed to ignore the Aesop and go full-time villain when it hits the fan.
Digimon Tamers subverted it as well. The Tamers lived like this at first, but when Calumon/Culumon was captured about midway into the story, they decided the "go to school" part was getting in the way and got permission (sort of) to take an indefinite leave.
This happens to Nagisa in Iczer One, but only after being nothing short of abducted by the eponymous protagonist to man the battle robot with her.
The premise of Yu-Gi-Oh! GX can be summarized as "Wake up, go to school, practice saving the world." By playing a children's card game. Which quickly turns much uglier than is typical. The school is also all about playing said card game. They might have overstepped this trope a bit.
In Yu Yu Hakusho, Yusuke flat out states that "school sucks" and that "this job" (basically, saving the world), is the only thing he's ever been good at. Probably justified in that Yusuke is extremely Book Dumb.
Sailor Moon. Maybe it's because Usagi/Serena totally sucks at/hates school (much like a lot of shoujo heroines).
Downplayed in the S season, where we see very little of the senshi at school, but nearly every episode starts with them studying for their exams (until they are inevitably distracted, either by the monster of the week or by some more mundane crisis).
Averted in Sailor Moon SuperS. They're all on summer break. During the Manga they've shifted to High School during this arc but it was pushed back an arc for the anime. The timeing pretty much ends up the same by the end though.
Mai HIME does this as well. With one exception, the girls are more concerned with boys, food, classes and part-time jobs (or if they're not in school, with their own affairs) than with destroying monsters, though they are more than happy to do so when they show up.
Later on in the series, this becomes increasingly averted as the battles of the HiMEs end up partly destroying the school, and most of them either drop out, disappear, or otherwise stop caring about what's going on in school because they're too busy trying to kill each other.
All but one of the Baka Rangers in Mahou Sensei Negima! are powerful fighters on Negi's team. Kaede and Ku Fei in particular were strong fighters even before the start of the series, yet their first major obstacle in the series was to receive adequate scores on their final exams.
Full Metal Panic practically runs on this. Most of Sōsuke's issues involve being unable to explain distressingly common situations and failing in epic fashion. In one episode, Kaname visits while Testarossa is in the shower and Sōsuke tries to explain why Kaname can't come in. Cue Testarossa sticking her head out asking if she can borrow clothes. He will generally sweat bullets in these situations — and yet, when he appears to be totally screwed in a military situation, he will either find a way out of it or somehow stay calm.
This is probably because, in those situations, he can't shoot anyone...
Subverted by Death Note. Or possibly played straight. It depends on your moral stance on Light's actions; he definitely believes he's playing it straight as he interweaves his increasingly meaningless social life with his genocide.
Shinichi, the hero of Parasyte. Somewhat deconstructed towards the end. Though he just barely manages to graduate, he flunks all his college entrance exams and is doomed to a life of poverty, and thus his future with Love Interest Satomi is far from certain.
In Bleach, Ichigo and his friends often have to make excuses, such as all simultaneously needing to go to the bathroom, in order to get out of class to deal with a Hollow in the area. However, as the threat of the Big Bad becomes paramount, pretty much all of them stop attending school for months to concentrate on their training.
In the first and last Eldran series, the kids actually had to go to school in order to save the world, as that's where the local Super Robot was normally kept.
Altered in Kekkaishi, where it's usually Wake up, save the world (school), go to school.
Invoked in the English release of the Shakugan no Shana manga, whose tagline is "Saving the world is easier than falling in love..."
There's also the (very high) possibility that the school Shinji et al attend is nothing more than yet another front for the Ancient Conspiracy and that all of the attendees there are Evangelion pilot candidates - at the very least, literally every student in Shinji's class is.
The Amazing Spider-Man may well have invented this trope, and Peter's constant struggles to keep his life on track while fighting crime shows up in almost every other incarnation of the series.
In fact, Stan Lee and Steve Ditko created Spider-Man around this very premise. They wanted a young superhero who, unlike the then-popular "sidekick" depiction of such a character, had to simultaneously deal with the social and emotional pressures of becoming an adult... and the parade of crazy costumed baddies.
Deconstructed in Scott McCloud's Zot The problems in Zot's world are on the surface bigger, but since it is ultimately a fantasy world the much smaller problems in Jenny's world are real problems and therefore more serious. The Deconstruction comes in the exploration of why high school life seems more serious than fighting super-villains.
Ostensibly, the Teen Titans — but the comics tend to ignore the characters' lives outside of super heroics to the point that Titans Tower occasionally seems like a weird super-teen commune.
Similarly, the X-Men mansion is technically supposed to be a "School for the Gifted" but only a couple of writers (among them Grant Morrison) really paid anything more than lip-service to the concept.
Let the Right One In doesn't deal with world-saving, but still exhibits signs of this trope. Oskar finds romancing his vampire (girl/boy)friend to be pretty easy◊; dealing with bullies◊, on the other hand, is much harder. If someone who wasn't familiar with the movie compared those two stills, they'd probably think that the all-too-human bullies are more likely to suck Oskar's blood.
Harry Potter got dangerously close from Goblet of Fire onwards, but Deathly Hallows abandoned many of the school based sub-plots to focus on the final confrontation with Voldemort. Justified in-universe in that Voldemort had an especially soft spot for the school - Hogwarts was probably the one of the only things in his life he came close to actually loving.
Young Wizards: Interestingly inverted in the fifth book, when the fact that Kit and Nita had a fight in the beginning and Kit wrote her name out of a wizardry they were going to do but didn't erase it allows him to save her life when she makes a Deal with the Devil to save her mother from terminal cancer by writing that version of her name back into a new wizardry and thus "resetting" her morally to the person who hadn't made any deals. Interestingly, although this works, reusing a name which also includes her age, physical characteristics, and every other piece of information about her in a setting where saying/writing it in the Speech makes it real doesn't have any other side effects at all.
The Alex Rider series deals with the potential real-world ramifications of just how much school one would have to miss to save the world. Not to mention what might happen to the school itself once the Big Bad knows exactly which school the teen- or Kid Hero goes to.
Animorphs made some use of this, but partially subverted it, as the characters soon find that fighting a guerrilla war against an alien conspiracy/invasion is physically and emotionally taxing, and their grades start to suffer. They lose all their friends not in the know, and stay in school just to keep up the pretense. Next to the enslavement of humanity, school seems kind of meaningless.
It also helped that they gained advanced ancient androids as allies who could take their place in school for long absences thanks to their advanced hologram technology.
Near the end of the series the war has escalated significantly, pressuring the kids to devote their entire lives to fighting. Jake notes that none of them get to school consistently anymore, to the point where Jake can't even recall what they're studying in class. They go on missions pretty much full-time, almost every day, and they rarely get more than a couple hours of sleep either.
The Demon Headmaster series of books. That crazy headmaster is always trying to take over the world (although he's not always a headmaster), and apparently this small group of children are the only ones who can ever spot it. But they still have to go to school and be home for tea.
Buffy the Vampire Slayer. That high school was infinitely worse than, or at best merely even with, fiends from hell was one of the show's basic concepts. Of course, this particular high school was built on a Hellmouth's centre, so the line between scholastic evil and demonic evil is blurred.
The season 3 finale ends thus (after having graduated and averted the apocalypse on the same day).
Oz: It's worth taking a moment just to think... we survived.
Buffy: It was one hell of a battle.
Oz: Not the battle. High school.
There's also the series finale. Right before the final battle, Buffy, Xander, and Willow start making plans for the day after, which involves shopping. Wake up, save the world, go shopping.
It is frequently lampshaded on the show that this trope is the reason Buffy is able to survive so long as a Slayer. Previous Slayers focused so much on training and fighting monsters that they never got to experience the normal lives and problems of a teenage girl. Without these experiences and connections the Slayers became careless and Death Seekers.
In Teen Wolf, for Scott, Stiles, Allison, and Isaac. Somewhat more murky on Lydia (who has no idea of the goings-on until the end of the second season) and Jackson (who's unknowingly the creature that's a threat).
In an episode of The Sarah Connor Chronicles, after John helped destroying a batch of coltan, a metal used to build terminators, he comes back home to do his homework for school.
Of course, he's since dropped out.
In Ace Lightning, Mark Hollander, a thirteen-year-old boy, unintentionally gets himself elected as the superheroes sidekick: this pretty much costs him his social life, including two girlfriends.
Which was lampshaded by the Red Ranger and Alpha the robot in a TV promo for the series. Red Ranger describes how he's a martial arts master and has girls drooling over him, then tells the audience: "What a cool gig. I save people, protect cities, fight monsters...." - only to be interrupted by a message from Alpha telling him that he has to do his homework.
Newer seasons is more like "Wake Up, Go to Work, Save the World."
Hex did a boarding school version, although it can be debated how much saving actually took place.
Likewise The Vampire Diaries where the world may not exactly be at stake, but there is a whole lot of saving and battling the forces of evil going on. They even do school reports on the subject of vampires!
Short lived British fantasy show Demons focused on the teenage heir to the Van Helsing title.
While technically not in school, Merlin does face both Big Bads and the trouble of dealing with teenage hormones, friendship issues and crushes.
The Sarah Jane Adventures— though it often seems like the kids have all the free time in the world, they do go to school and a number of episodes either take place or have a number of scenes set in school.
Kamen Rider Fourze: Conveniently the school just happens to located at a major source of cosmic energy. Inconveniently this is because it was founded by the series Big Bad to advance his plans and several members of the staff are his subordinates who are responsible for turning students into the Monster of the Week.
Mage: The Ascension mentioned this when describing how the Game Master could use character backgrounds to make the game more interesting. Fighting the Ancient Conspiracy of wizard-scientists trying to eradicate magic from the world, the bunch of Evil Sotcerers trying to destroy everything that has ever existed, the mad mages whose very existence warps reality, and the odd elder horror from beyond the stars gets a lot harder when your mom is going to kill you for staying out past curfew.
Persona 3 bases both its storyline and gameplay around this concept, requiring you to balance fighting demons with day-to-day activities to succeed. It also subverts it, to an extent — the characters frequently admit that they find fighting Shadows much more difficult than dealing with school.
Most of the world-saving goes on AT school (albeit a supernaturally altered one.)
It also plays with this a bit as unlike most of the examples here, the actual world-saving happens not only at the school but also at a specific (and supernatural) time of day that can be planned around, this allows for the balancing aspect of the trope to be a touch easier.
Persona 4 does much the same thing, though in this case it's Wake Up, Go To School, Catch a Serial Killer. Although as you find out later, catching the Serial Killer is part of saving the town, so it's actually both.
In fact, amusingly enough, you can only go into the TV (to save the world) after school every day. You literally have the schedule of Wake up, Go to school, Save the world.
Both Mega Man Battle Network and Mega Man Star Force play this one straight by having the main protagonist be a fifth/sixth-grader who also happens to either own or be the most powerful being on the internet. Throw in a villain of the week or fifty and you're set to go.
In Star Force's case, it takes a lot of work from Echo Ridge's local Class Representative/#1 Mega Man fangirl to get the hero to "GO TO SCHOOL" (emphasis hers).
Humorously averted with Riki and Mami in Bangai-O, who spend the entire game travelling through the galaxy to defeat the Cosmo Gang. This results in Riki's health teacher tracking him down with a stolen mech.
Lollipop Chainsaw begins as just another school-day for our heroine Juliet Starling, but soon escalates to a grand adventure of saving the world from a Zombie Apocalypse.
Touched on in Gunnerkrigg Court: Emotionless Girl Antimony has no problem talking with ghosts or judo-flipping the class bully, but she has difficulty making normal conversation with her classmates. She's gotten better since opening up to her friend, Kat.
Several of El Goonish Shive's characters have struggled with issues such as coming to terms with their sexual orientations and standing up against unfair school policy. Meanwhile, there are interdimensional conflicts going on that directly relate to the central cast, but those only come up in conversation as direct responses to the events happening, and even then they don't last very long.
Used in Angel Moxie, and some teachers are more than meets the eye.
School administrator: Look, I don't care if you girls are mutants or Supergirls or Wonder Women or what, you do not get out of having your parents called when you bring God only knows what kind of monsters to school with you. Not to mention skipping out of class.
To the kids attending the Hyperion Academy or enrolled at the Venture Institute, beating up bad guys is nothing compared to life in High School. Booster is wondering what his friends on the soccer team will think if he tells them he is gay. Barnstormer likes that he's popular, but is worried that he won't pass College Algebra. Donnybrook not only thinks the Headmaster of the Hyperion Academy is a butthead, he wonders how in the heck he can get a new Code Name, because... Donnybrook? Really? Calendar is worrying her head off over whether she should go to Harvard or Cambridge. Bouncer hates being thought of as "The Fat Kid", but worries that his powers depend on being as heavy as he is. Double Trouble is beginning to suspect that the two girls he is using his duplication power to date simultaneously have learned of each other's existence. Typical teenage angst, really.
LessThanThree Comics' Brat Pack are an example of this trope, being made up of high school students, who also save the world on a regular basis.
Pretty much standard for the superpowered kids of Team Kimba at the Super Hero School Whateley Academy in the Whateley Universe. They even have specific scenes or story lines focused on classwork: Tennyo has trouble with math; Chaka is struggling with English; Shroud doesn't get some of the aspects of physics that her body apparently violates; Phase is bored stiff in her Intro to Superpowers class because the professor is so tedious...
The younger heroes in The Descendants attend high school, but are rarely shown in classes. Meanwhile, the students in the Spin-Off, Liedecker Institute are shown almost exclusively in school, but haven't performed any heroics yet.
Worm starts out like this in the very beginning and then deconstructs it hard. Taylor starts out using the superhero life as an escape from her miserable high school experience, and later on stops going to school at all. Brian, the oldest of the team, already graduated and takes care of his younger sister, Lisa tested out early using her power to cheat on the GED test, and Alec and Rachel never went to school regularly due to their unstable, extremely dysfunctional home lives. When their hometown is ravaged by monsters and supervillains, all the schools are pretty much shut down, and school doesn't really seem to matter anymore when everyone is just struggling to survive on a day-to-day basis.
Popularized within American cartoons by Kim Possible. Even before the series rolled out, TV previews summarized the premise of the series as a girl who finds saving a hijacked space station far easier than asking a cute boy out, a problem she faces in the very first episode to be broadcast ("Crush"). Highlighted within the series itself in The Movie, when Kim responds to Drakken's taunting about her social drama with, "You're right, Drakken. Boys? Dating? It's hard. But this? Is easy!" At which point she slugs him.
A variation on the theme shows up in the driver's-ed episode — the teenage rite of passage of learning to drive a car was a lot harder for her than flying a jet pack or landing a space shuttle.
Increasingly deconstructed, as the constant attacks leave little room for studying or staying in class. Particularly from the second season onward, where the Return to the Past is used less and less, as each use increases the memory of the supercomputer, and by extension XANA's power.
X-Men: Evolution Though in the early seasons this got a bit of flak for too much school, not enough world saving (less "how can I use my powers to help people?" and more "how do I stop people from knowing I have powers?" essentially). The outing of mutants "fixed" that though, bringing in more action and less wangst.
That problem with not enough action stemmed form a Broken Aesop for the first two seasons: Xavier preached about how humans needed to learn to accept mutants. Humanity does not know mutants exist (one assumes there are no other superheroes in the world, but then they had a flashback with Captain America....), so how can they learn to accept them?
Parodied, mocked, and even averted in Invader Zim. The eponymous character may be an evil green alien attempting to bring about the annihilation of the human race, but of course he spends plenty of time pretending to be an elementary student ("It's a... skin condition). Almost, because the good guy, Dib, a classmate of Zim, spends much of his screentime either ostracized in school, dealing with his cruel little sister, and... trying to save expose/save the world to/from supernatural threats.
Who could forget Batman Beyond, since this was a major dilemma for the Batman-who's-still-in-high-school Terry McGinnis. A lot of sub-plots in the episodes revolve around this, and even a few major plots do, like when Terry takes his schoolwork on the job in "The Eggbaby". Rather coincidentally, a lot of the villains and problems in the show ALSO come out of Terry's high school.
What makes this notable is that Bruce allowed Dick/Jason/Tim time off to go to school, do homework etc since they were Robin. But since Terry is Batman, he is expected to be on call 24/7.
For Winx Club, this would be the girls' daily routine during seasons 1 through 3 as they attend Alfea. The Winx graduate during The Secret of the Lost Kingdom, so they no longer apply for this after that.
Transformers Prime: Though its more like "Wake Up, Go To School, Help The Giant Robots Save The World And Maintain Their Cover." Jack takes it the most seriously, though Raf is probably the most helpful. Miko can get a little carried away with her excitement.
Averted in Iron Man: Armored Adventures. Tony Stark has very little interest in keeping up with his school life and would much rather be working on new armour in the lab or flying around, saving the world. Nevertheless, he seems to have made friends pretty easily despite never being in a high school before and taking very lengthy "bathroom breaks" seemingly every other day.
Sym-Bionic Titan, although they started going to school around the same time they started saving the world, the former is more of disguising themselves as Earthlings.