People — particularly those with authority, be it moral or political — are expected to act in a manner in accordance with the ideals they espouse. That is to say they should practice what they preach.
Not all who fail to live up to these standards are hypocrites. Some people may fervently and honestly believe what they say is right and good... they just don't have the moral strength or willpower to consistently live up to their own high standards. These aren't hypocrites. A hypocrite, by definition, only pretends to believe what he preaches, although the more self-aware among them may occasionally acknowledge (and/or attempt to justify) their situation with a Hypocrisy Nod.
Frequently, they will be found out. Be it an Engineered Public Confession or through investigation. The Hero may have the choice of either exposing them as a fraud and discrediting them before their followers, or keeping their secret and blackmailing them into cleaning up their act or helping in another matter. How this turns out depends on how sympathetic or Jerk Ass they are, and how humanizing their "vice" is (a diet guru eating donuts is probably ok, an eco-businessman clear cutting forests probably isn't, a moral crusader molesting children definitely isn't). If a villain finds a friend of the hero's Fatal Flaw this way (or worse, the hero's own), they might use Flaw Exploitation to torment and control them.
If found out and/or exposed, the hypocrite will have the chance to mend their ways and do a Heel Face Turn in one of two forms. Either loosening their standards and living a simpler but more honest and happier life, or tightening their belt and actually live up to their ideal. Failure to do either is usually enough for either a mental breakdown or a full on Face Heel Turn as they reject their morality and embrace their vice.
Alternatively, because they are feigning what they claim to be, they may find they are Becoming the Mask.
Heroes are often accused of hypocrisy by villains who want to believe they're Not So Different and brag At Least I Admit It. Heroes who actually are hypocrites tend to hear "What the Hell, Hero?" quite a lot. (Unless they don't.)
Marion Keeny in Scarecrow: Year One calls her daughter a whore for having a child out of wedlock and suggests burying the (living) baby in the backyard, but the fact that her and her daughter's surnames are the same as that of Marion's mother implies that she may not have been married either.
Or they just kept their maiden names.
In Salvation Run, Gorilla Grodd mocks Monsieur Mallah's claims that they are similar and derides him as the uplifted pet of a Mad Scientist. Grodd conveniently fails to mention that he was uplifted by an alien Mad Scientist.
In Batman: Heart of Hush, Hush mocks Batman's crime-fighting career as a sign of his inability to move on from his past. This is pretty rich considering that his own vendetta against Bruce stems from a grudge he's held since childhood for something that wasn't even Bruce's fault.
In Blackest Night, Captain Cold justifies killing the second Captain Boomerang by claiming "Rogues don't kill women and children." Trouble is the lineup at the time includes the second Mirror Master (who definitely killed Rachel Rathaway as well as the Rathaways' female housekeeper), the second Trickster (unless all the homeless people he blew up happened to be adult men), and Heat Wave (again, unless he somehow only caught adult men in his numerous acts of arson).
Just to add some examples: one story features him telling Zipi and Zape off because he thought they were smoking cigarettes (they weren't), explaining how unhealthy they are. He doesn't mind that he's almost always smoking himself.
Don Pantuflo often tells Zipi and Zape off because of their bad marks on school, telling them he always got A. One story, however, revealed he never achieved more than a B. Although whether this is canonical is questionable, the fact is that Pantuflo is always depicted as having trouble at the time of helping Zipi and Zape with their homework.
In the Jackie Chan Adventures fic Queen Of All Oni, Ikazuki looks down on humanity, even giving Tohru a Hannibal Lecture about it, but the speech is shot down by Tohru, pointing out the Oni General's hypocrisy: Ikazuki was defeated by humans, and even now needs a human body (not to mention he was wearing one currently) to do, well, ANY PHYSICAL ACTIVITY.
Not to mention his disdain towards Tohru's continuing concern for Jade, even though he has strong loyalty towards Tarakudo.
Many "Alternate Universe" Harry Potter fanfics have depicted Harry being Sorted into Slytherin instead of Gryffindor. In almost every single one of them, Harry becomes a pureblood bigot, subscribing to and even championing Slytherin House's philosophy condemning Muggles and Muggle-born wizards as inferior... despite the fact that Harry's own mother, who sacrificed her life to save Harry's, was Muggle-born.
In the opening scene of the Clannad fanfic An End To All Things, Okazaki advises Furukawa to not live in the past. What was he doing shortly before he told her that? Reliving a memory.
Death Note Equestria: Twilight says the Second Kira disgusts her for killing innocents, even foals (like Sweetie Belle). This in spite of having killed plenty of innocents to protect her own hide, including a reporter not much older than Sweetie Belle.
Lereal Belsai of Hivefled thinks of himself as a devout Sufferist, but instead of thinking the hemospectrum ranking should be removed, he wants it reversed and the coldbloods enslaved in turn, hating them to the point that he dismisses Gamzee as a spy out of hand despite the obvious marks of torture on him, and demands that a ragtag collection of Child Soldiers try to take on the entire empire. He's been described by a reader as the Malcolm X to the Sufferer's Martin Luther King.
The Prayer Warriors have too many examples to list here, but the one that best notes their tendency to fall into this is when Jerry says "Killing a Christian is a sin", and in the next paragraph, when Thalia Grace comes up to him repenting her sins, Jerry kills her, believing that she deserves to die if she's lying and if she's telling the truth, she will die a Christian death.
Prison Island Break: Silver finds himself betraying his friends and denouncing God in order to survive in prison.
Hypocrisy Nod: He's well aware of his hypocrisy and is disgusted with himself.
The 'Social Justice Sally' meme targets hypocritical and self-righteous social justice 'activists' who just act exactly the same as the kind of bullies and bigots they claim to oppose.
Jay-Z put "Death of Auto-Tune (D.O.A.)" on an album with several Auto-Tuned hooks. Jeez.
Several songs on Michael Jackson's HIStory have him railing against injustice and persecution, both of people at large and himself specifically, as he was being hounded by the mass media at the time over child molestation allegations. In "Childhood", a key lyric is "Before you judge me/Try hard to love me" — i.e., don't be so quick to condemn based on assumptions. On the same album is "D.S.", a song about Tom Sneddon, the district attorney who ultimately headed up both of the investigations of Jackson's behavior with children. Sample lyric: "You think he brother with the KKK?/I know his mother never taught him right anyway"...what was that about judging people again, Mike?
In the film Moonwalker, he included the song "Leave Me Alone" which railed against the public attention he was always hounded with, in the middle of a film that can best be described as a 90 minute commercial for the Michael Jackson brand.
The folksinger Phil Ochs loved to ridicule this trope from any side of the political spectrum. "Draft Dodger Rag" is about a red-blooded conservative who's all for that war in Vietnam, so long as he doesn't have to go himself, while "Love Me, I'm A Liberal," is about someone who pays lip service to every left-wing cause until it becomes dangerous, distasteful or personally uncomfortable. ("The people of old Mississippi/ should all hang their heads in shame,/I can't understand how their minds work./ What's the matter, don't they watch Les Crane?/But if you ask me to bus my children/ I hope the cops take down your name ....")
Metallica got famous by tape trading their cassette demo "No Life Til Leather", which landed them a record deal. They also were known to allow fans to record, visual and audio, their sets and circulate them freely. But the moment they got big and started making real money off their music they sue Napster over doing the same things that got their fiercely loyal fan base to begin with.
Goblin Hollow: Gothchilde complains that people are hypocritical fakers — while claiming to be a 300-year-old vampire.
Living with Insanity had an arc where Alice had to deal with customers who were rude, overly demanding and blamed her for things she couldn't control. When David takes her to a cafe after work, she does this.
In Mortifer, it's explained that demons get more powerful from certain emotions or lifestyles related to their power. While all of them become more powerful as they lose their grip on reality, Zebidiah, as a shapeshifting demon, becomes more powerful the more hypocritical he is. Which is why he works as a priest despite being a demon.
Another example is Redcloak saying that paladins are unnatural abominations due to their magical lack of fear. He even refers to himself as a "100% all-natural goblin" in his Hannibal Lecture. He conveniently leaves out the fact that he's wearing a magical artefact that has prolonged his natural lifespan by decades.
At the climax of Suburban Knights the Big BadMalachite, after spending the series killingpeople for being overly reliant on technology (and unlike most killing on TGWTG.com shows, his actions are NOT played for laughs) and in the middle of declaring his intention to destroy all technology, is interrupted by an utterly trivial call on his iPhone. He is promptly called out on for this, and initially responds by trying to deny that an iPhone counts as technology.