The Guardians of the Universe in the Green Lantern mythos frequently fill this role, most recently in the "Blackest Night" event, where all save two refuse to believe in the prophecy and the coming of the War of Light until it is too late.
The Jedi Council in the Star Wars prequels, particularly the first one. Even after they recognize the problem, they are hard-pressed to not act like idiots. In fact, they seem to discover there is a growing conspiracy in the Republic in each film, seemingly at the same time Palpatine has another rise in power, but choose to ignore all the signs or put two and two together until the third film.
Count Dooku leaves the Jedi because of this, among other reasons. When he later turns up as the leader of an enemy movement, the Jedi and Chancellor Palpatine STILL sit on their butts until the very last second. Palpatine has an excuse (he's the Big Bad and Dooku is secretly his minion). The Jedi don't.
Chancellor Valorum is this, too.
Amidala: I was not elected to watch my people suffer and die while you discuss this invasion in a committee!
Padme qualifies as well. She leads a cadre of senators who want to defeat the proposed Republic bill to create a standing army. It really isn't her pacifism, but the fact that, given who the Republic is dealing with, she should know better. The Trade Federation quickly curbstomped her planet only ten years before and they're the ones bankrolling Dooku, along with a dozen other similar organizations.
King Théoden flirts with being this briefly in the film version of Lord of the Rings: The Return of the King, initially refusing to help Gondor, until the beacons are lit and he decides to send in The Cavalry. His reticence is justified however, since Gondor had increasingly refused to aid them in recent years, something he calls Aragorn out on when he brings up their Binding Ancient Treaty. It's implied his decision is partially because he wanted to make a point that at least Rohan still had the honour to live up to their commitments.
The Archon in Immortals is sure that he can negotiate with King Hyperion, right until the moment where Hyperion kills him.
David Weber: The liberal and pro-peace political parties in any of his novels are written as naïve people who believe in the "peace at any cost" philosophy.
But not the Bahzell series. There the pro-peace liberals are the good guys.
Cornelius Fudge in the Harry Potter books and films. He ended up getting the sack after the true nature of the threat became apparent, lingering, like Chamberlain, as an advisor to his successor. It's possible that Fudge was intended to sit in for Chamberlain, who had a similar approach to the rise of Hitler until From Bad to Worse.
Brilliantly parodied in A Very Potter Musical when Voldemort breaks into Fudge's office to take over the Ministry of Magic:
Voldemort: Cornelius Fudge, the Minister of Magic! Fudge: I still don't believe you're back. Voldemort: Believe this, Fudge! Avada Kedavra! Fudge: Oh, heart attack... surely. [dies]
Another example is found in Lego Harry Potter: The whole stance of the guy is summarized by him putting his hands over his ears and going "bla bla bla"
The Board of Trustees of the First Foundation in the first part of the novel Foundation of Isaac Asimov. They represent a specific part of the decadence of the galactic empire: the excessive reliance on a greater authority.
Prince (later King) Meurig of Gwent in The Warlord Chronicles. He starts out by convincing his fellow Gwentians to back out of the war with Powys, which any realist would see would quickly lead to Dumnonia becoming a Powysian puppet-state, thus surrounding Gwent on three sides. In the second book, while king, refuses to intervene in a civil war in Dumnonia in support of Arthur, potentially leaving governance in the hands of a group of Saxon-backed conspirators. In the third, he refuses to join with the other British nations in resisting the Saxon invasion of Dumnonia, which proves a bridge too far: his father Tewdric returns to the throne briefly to resist the invasion. The series' framing device makes clear that Meurig's kingdom was within his lifetime mostly conquered by the Saxons.
Mr. Desjardins, the Chief Lector of the House of Life in The Kane Chronicles, spends most of the first book either ignoring or seeming not to care that Set is about to destroy the world, being more concerned with killing the god-hosting heroes. He eventually graduates to Divided We Fall.
The Governance Kernel of the Sentry Coalition, in the Star Trek: Titan novel Synthesis. They refuse to listen to the warnings of SecondGen White-Blue and FirstGen Zero-Three regarding an impending incursion of massive proportions by the Null. White-Blue is frequently dismissed as unduly alarmist.
The Star Wars Expanded Universe gives us the Fallanassi, who are such pacifists that, even when billions of people are dying, they refuse to take up arms.
The Pre War Presidential Administration in World War Z, who ignored the solutions to preventing the looming global zombie outbreak, instead performing a few token solutions that ranged from unhelpful to genuinely harmful (i.e. allowing a bogus vaccine to pass through the FCCC). When interviewing the former Chief of Staff, Grover Carlson, Max Brooks calls him out on this.
Every. Single. Character with access to military resources in A Song of Ice and Fire. After thousands of years, the Others have returned and are raising an army of the dead beyond the Wall? What imaginations the men of the Night's Watch have. Zombies or not, there's definitely an invasion underway and the Wall is hopelessly underdefended? Great, that should keep the King in the North busy and out of the fighting for the Iron Throne.
The White Council in The Dresden Files. The Merlin and his people want to appease the Red Court (by offering them Harry's head), when it's fairly clear that the vampires want a war. They also completely deny the possibly of a rogue faction within the Council, despite mounting evidence. At least some of this behavior was probably due to a traitor using mind magic to subtly influence their decisions.
Zeus/Jupiter from Percy Jackson and the Olympians and The Heroes of Olympus is aware of the rising threat of the Titans and Giants, but tries to deny and ignore it until his fellow gods force him to confront it with irrefutable evidence. By the time he decides to take action is is too late and he has to rely on mortal heroes which is partially what galls him in the first place.
President David Palmer on 24 is accused of being this in-story by his Vice President in season two, though he turns out to be right.
Used many times in all incarnations of Star Trek. Chamberlains can be found liberally sprinkled among Starfleet's admiralty, the Federation's civilian leadership, and the councils of many alien worlds. The stick they wield is the Prime Directive, originally drafted to prevent the Federation from imposing its will on defenseless alien species, now ironically aiding and abetting the very same.
Discussed in the popular Deep Space Nine episode "In the Pale Moonlght." Sisko sells his best arguments for why the Romulans should stop aiding the enemy, while Dax role-plays as a Romulan proconsul. If it goes this badly in rehearsal, just imagine how unpleasant the real deal would be.
Dax(as Romulan): From where I'm sitting, the Dominion isn't a threat to me. I have a non-aggression pact with them. They're my allies. Sisko: You're not going to put your faith in some piece of paper, are you? Dax: Not at all: I've been watching them very closely since the beginning of the war, and so far, they've kept their part of the bargain... Sisko: They're violating your territory almost every day! What kind of an ally is that?! Dax: So they're crossing my backyard to give the Federation a bloody nose. I can't say that makes me very sad. Sisko:(getting incensed) You can't be naive enough to think that the Dominion is going to stop with the Federation! When they're finished with us, they're coming after you! Dax: That's speculation— Sisko: The Founders see it as their sacred duty to bring order to the galaxy — Their order! Do you think they'll sit idly by while you keep your chaotic empire right next to their perfect order? No! If you watch us go under, then what you're really doing is signing your own death warrant!
In the second-season finale, as the station is gearing up for possible conflict with the newly (re-)military expansionist Centauri, a high-level diplomat comes to the station from Earth...and reveals that he's there to make a non-aggression treaty with the Centauri. He even says "peace in our time." However, while he was serious in his intent, he was chosen for this task by superiors who were working with the Shadows (who didn't want their pawns fighting).
Early in the fourth season, the Drazi and Hyach ambassadors who want to prevent Delenn and her few remaining supporters from continuing attacks on the Shadows. They believe that if the Army of Light doesn't antagonize the Shadows, they might well go back to sleep for another thousand years, but if Delenn and company press on, it will drag everyone down into the abyss. As is usual with this trope, the Drazi and Hyach ambassadors aren't antagonists... they're just wrong.
Happens a few times in Stargate SG-1, notably Kinsey and Woolsey (though the latter got character development). Repeatedly, when there is an Obstructive Bureaucrat, their problem is that they don't think that the enemy of the season is a real threat.
It gets worse after the introduction of the I.O.A., who insist on inflicting ridiculous policies on SGC personnel, all whilst ignoring the various major threats that are posed to annihilate Earth or the galaxy at any given moment. For the most part, the SGC personnel seem to consider the I.O.A. to be a joke organisation and are well aware that it's members are such slaves to PR, they'd never want to be seen complaining about Earth being saved because they were ignored.
In the original Battlestar Galactica, the President worries about offending the Cylons, brushing off some very concerning warnings, only to lead his people into an ambush that nearly accomplishes the genocide of his entire people. The chief peace broker was working for the Cylons the whole time.
The former Trope Namer is referred to in an episode of Seinfeld when Jerry is assuring George that an executive at NBC won't cancel their pilot just because Kramer threw up on her.
Jerry: Vomit is not a deal breaker. [...] If Hitler had vomited on Chamberlain, Chamberlain still would have given him Czechoslovakia. George: Chamberlain? You could stick his head in the toilet, he still would have given you half of Europe!
Veronica Mars can't seem to meet an authority figure who isn't a complete horse's ass.
Referenced in Civilization IV, where after meeting a rival civilization for the first time and you don't immediately declare war on them, your response is "There shall be peace in our time!" It's a very rare game indeed when such optimism isn't proven to be misplaced.
Much like the later council in the Star Wars prequels, the plot of the Knights of the Old Republic series is kick-started by the Jedi Council being content to sit and wait as Mandalorian armies ravage the Republic, with a number of Jedi going against their wishes and following Revan off to war. What's left of the council continues this sort of behaviour in the second game - when the Exile find them on your own, they're reasonable enough, but bring them all back together on Dantooine and they immediately turn on the Exile, out of fear because the Exile became a Force black hole after Malachor V, declaring them to be Sith and contemplate having the Exile killed.
In Star Wars: The Old Republic, the Trooper storyline has you go after a highly competent and ruthless Sith general. After a lot of hard work, luck and Heroic Sacrifice you succeed in capturing him. You are given a medal and told that the Republic has traded the prisoner back to the Empire for some trivial concessions. There is an undeclared war going on with the Sith Empire that the Republic is losing but the Senate sees no reason to be concerned and things that they can still negotiate with the Sith.
In the Mass Effect series, the Council has this attitude towards the Reapers despite Shepard repeatedly telling them that they're coming, especially Sparatus, and Shepard has the opportunity to not only call them out on it, but can even refuse to save them in the first game, not that their replacements are any better. All three quickly do a 180 in Mass Effect 3 when the Reapers finally arrive, and amusingly turian Sparatus is the most supportive of the three. Salarian Valern requires Shepard to rescue him from Cerberus, and the supposedly reasonable asari Tevos first states that, "The cruel and unfortunate truth is that while the Reapers focus on Earth, we can prepare and regroup," then refuses to give Shepard a vital clue until it is already too late. This ends up resulting in Laser-Guided Karma when Thessia ends up being the only Council homeworld to completely fall to the Reapers.
Lampshaded during the Citadel coup attempt. When you tell them Udina sold them out to Cerberus; Tevos admits that every time they've doubted you before, i.e. every time you've spoken except when you proved Saren's treachery, it hasn't ended well for them.
Grand Cleric Elthina of the Kirkwall Chantry in Dragon Age II. You can warn her about anti-Qunari fanatics abusing her seal, but she doesn't act against them until the Viscount's son is murdered and violence is all but inevitable. Her refusal to take a stand in the Mage-Templar conflict leads to both sides becoming radicalized - the Templars turn Kirkwall into a borderline police state, while cornered mages turn to blood magic in droves. Anders gets so fed up that he blows up the entire Chantry with her in it, forcing the issue.
Viscount Dumar also tends to take a very quiet route in regards to the Qunari. He clearly doesn't like them and doesn't like his son hanging around with them, but his only actions amount to sending Hawke to try and appease them. Part of the problem is that the Viscount doesn't want to do anything lest he incur the wrath of the nobles, but neither can he take a direct approach as the Arishok refuses to say anything to him other than "Begone", forcing him to rely on Hawke, as the only person in Kirkwall the Arishok considers worthy of his attention. After the murder of his son by Chantry zealots hoping to frame the Qunari, he stops caring altogether, leading to the situation deteriorating rapidly, the Qunari beginning a military coup of the city and his subsequent execution by the Arishok.
The Argent Crusade and Cenarion Circle with regard to the Horde trampling on their stated purposes in Worldof Warcraft.
In Sluggy Freelance the leaders of the Dimension of Lame jump head-on into this trope when they're attacked by Demonic Invaders. Their entire defense strategy consists of two phases: first, ask the demons to stop killing them really nicely; second, if the first phase doesn't work, ask the demons to stop killing them really nicely a few million times. Sending them fruit baskets is optional, but a plus.
Of course, this is somewhat justified by it being an entire dimension of Actual Pacifists, to the point where the most violently psychotic person in the entire dimension is horrified when he stubs a demon's toe. And when two people come to the reluctant conclusion that fighting might be necessary, they implement that plan by arguing with each other in an attempt to scare away the demons.
Credenza: What you're telling me is incredibly frustrating and I don't like hearing it. Isaac: That's why I brought it to your attention, sir. Credenza: No, this is what's frustrating. That you are bringing it to my attention.
Long Feng, the Evil Chancellor of Ba-Sing-Se, has elements of this. While he's clearly aware of the war with the Fire Nation, he seems more interested in keeping it under wraps than actually doing anything about them, and he spends far more time scheming against the Avatar, the world's last realistic hope, than against his enemies.
Though he's not a completely straight example, as he is portrayed as outright evil, and he's very ruthless in enforcing his own power- he just doesn't want to jeopardize it.
The leadership of the Northern Water Tribe also falls under this, having spent 85 years of a 100-year world war doing absolutely nothing following an earlier skirmish with the Fire Nation. They only rouse themselves to fight off a second invasion of their fortress-city, and after that they resume doing absolutely nothing for the rest of the series (contributing no forces to the last-chance invasion on the day of Black Sun, for example), although quite a few of them, including their greatest waterbender, were busy rebuilding the Southern Water Tribe.
On a much smaller scale than the above, the leader of the village in "Avatar Day" seems more interested in executing the Avatar for crimes committed in a past life than in helping the Avatar save the world, until Fire Nation soldiers arrive on his doorstep.
All in all, the reason the Two Nations were losing the war became increasingly clear over the course of the series.
A Subverted example from "Return to Omashu" would be King Bumi who when Fire Nation troops attacked immediately surrendered and simply cackled about doing nothing, but subverted in that he was simply waiting for another moment to strike.
Also subverted with the Earth King. He was practicing Head In The Sand Management because he genuinely didn't know about the war (thanks to Long Feng). When he discovers it, he laments how long it took for him to act and immediately works with Aang to help coordinate an invasion plan.
Raiko from the sequel series The Legend of Korra arguably counts as this. He refuses to help to Korra free the Southern Water Tribe from the Northern Water Tribe because it's something that does not directly involve the Republic, and even when told later on of Unalaq's evil plan to free the terrible Vaatu from his prison he still declines because he feels it's better to simply try to prepare for the worst-case-scenario and ready Republic City's forces to try and weather the storm. Naturally, due to his lack of help, Korra fails to prevent Vaatu's escape, and Republic City's defenses are utterly useless against the hybrid form of Unalaq/Vaatu, who would have destroyed the City had Korra not arrived in time to stop them.