Kirk and friends steal the Enterprise and defy Starfleet orders to not return to the Genesis planet in order to rescue Spock.
Sulu: The word, sir? Kirk: The word is "No". I am therefore going anyway.
Kirk's eventual "punishment" for this is to be demoted in rank to Captain - which is actually what he had wanted all along, and is Starfleet's way of rewarding him for what turned out to be heroic actions. As it turned out, they were glad to admit he was right.
Captain Jean-Luc Picard disobeys the orders of Starfleet Command and goes to the front lines to engage the Borg.
Picard: What I am about to do is a direct violation of our orders. If anyone objects, please do so now. It will be noted in my log. Data: Captain, I believe I speak for everyone here, sir, when I say... "tohellwith our orders".
Which creates a sort-of Call Back to the previous entry, given that Data is an android who's only recently begun to seriously express emotions and is thus very similar to Spock, making both their contexts both unusual and poignant.
Star Trek: Insurrection, the entire plot revolved around this trope so much that their rebelling against the rules is actually part of the title. The thing which sets the plot off is Data, who was assigned on what he thought was a survey mission, and attacked to keep the truth hidden when he discovered what was really going on. This damage kicked him into a kind of basic mode of functioning, probably designed to keep him from being used as a weapon, wherein his program directed him to do the right thing, regardless of whatever else was going on. Essentially, he was programmed with a Screw the Rules, I'm Doing What's Right! default mode.
Star Trek Into Darkness starts with Kirk and crew violating the Prime Directive to save an alien tribe from a volcano (possibly, it's a little clear if that was the big violation, or if letting the natives see the Enterprise during a rescue of Spock. Either way, writing a misleading report to cover it up afterwards wasn't Kirks best decision). Subverted in that Kirk ends up losing his command as a result. This is later inverted when Admiral Marcus gives Kirk the Enterprise back and orders him to fire a payload of advanced photon torpedoes from a distance at John Harrison, who was in hiding in Klingon Space( as part of Marcus' ploy to instigate a war between the Federation and the Klingons). After much soul-searching, Kirk decides "Screw the Admiral's questionable orders, I'm doing what's right" and informs his crew that they are going to arrest Harrison and bring him to Earth to stand trial.
In Transformers: Revenge of the Fallen, Lennox and the other NEST soldiers fly off not only to deliver Optimus Prime to Egypt, but actually take the Obstructive Bureaucrat with them so they can push him out of the plane on the way (with the Parachute, unfortunately). The latter was presumably because it would take him longer to rat them out than if they left him at the base. And because Lennox was having way too much fun screwing with the guy.
In the film version of The Running Man, Ben Richards is ordered to fire upon a food riot. When he refuses, the crew overpowers him and carries out the order. Richards is then blamed by the state and becomes known as the "Butcher of Bakersfield".
In The Matrix Reloaded, Neo realises Trinity will be killed by an Agent, and insists she stays out of the matrix; however, despite his pleading, Trinity states "[She refuses] to sit and watch [Neo] die", and does so anyway.
Crimson Tide is all about this. Lt. Commander Hunter (played by Denzel Washington) actually commits mutiny and seizes control of the USS Alabama in the name of preventing nuclear war. Though Hunter insists throughout that it was not a mutiny, he did everything "by the book". It was the Captain who disobeyed proper procedure, by not holding the launch countdown pending retrieval of the message, and attempting to relieve Hunter for fulfilling his role, the very reason why there's two sets of keys. As far as Hunter sees it, his actions were all Lawful as well as Good.
In I, Robot: Sonny, a advanced robot who is able to think independently of the three laws, agrees that the actions of the main villain are perfectly rational and that their logic is sound in accordance with the laws of robotics; however, he chooses to go against their plan because it "just seems too heartless."
In Hellboy II, Liz Sherman and Abe Sapien go against orders to take a dying Hellboy to Prince Nuada's realm in order to save his life. Johann Kraus intercepts them, seemingly intending to either reason with them or arrest them for disobeying orders, but instead joins them.
G.I. Joe: The Rise of Cobra: General Hawk first subverted his orders by telling the team that they could violate the spirit of the orders without technically violating the letter and later launched an unsanctioned attack on Cobra's Arctic base after the organization was ordered disbanded.
Street Fighter: "Troopers, I just received new orders. Our superiors say the war is cancelled. We can all go home. Bison is getting paid off for his crimes, and our friends who have died here will have died for nothing. But, we can all go home. Meanwhile, ideals like peace, freedom, and justice, they get packed up. But, we can all go home. Well, I'm not going home. I'm gonna get on my boat, and I'm going up river, and I'm going to kick that son of a bitch Bison's ass so hard that the next Bison wannabe is gonna feel it! Now, who wants to go home... and who wants to go with me?"
In the video game adaptation, failing to complete the assault on the compound gets Guile court-martialed.
National Treasure is all about a guy who steals the Declaration of Independence so someone else can't. In the sequel, he kidnaps the President, but unlike the first movie, it has absolutely nothing to do with "saving the country" or anything; Gates just wants to find El Dorado and clear Thomas Gates' name of treason.
Has a scene where Sergeant Zim is arguing with his superior officer to let him join the war. Being a boot camp instructor, he won't get anywhere near the front lines unless he "busts himself back to a Private". Rico bursts in, asking Zim to cancel his resignation so that he too can join the war effort. Zim shows him the resignation documents, and after a silent nod from his superior officer, rips them up and thus gives both of them what they want.
The end of movie shows that Zim did end up busting himself down to private, though that's probably a subversion, since he was actually following the rules in that case. And it turns out to be the best decision as it was Zim who captured the Brain Bug.
The primary reason Jason keeps the evidence from the police in Mystery Team.
In Tears of the Sun, the SEAL Team engages the Nigerian rebels after watching the rebels massacre a village, not to mention trying to extract as many indigenous refugees from the conflict zone as possible, against direct orders from their command center.
Steve Rogers' first real action in World War II is when he rescues 400 soldiers from HYDRA against Colonel Philips' orders. Subverted to a degree when, after returning the men to their base successfully, Steve voluntarily surrendered himself for disciplinary action, only to have Phillips say, "I don't think that will be necessary."
Agent Peggy Carter decides to help Steve to get into the HYDRA base, at the risk of her career. Likewise with Howard Stark, who flew the plane into enemy lines.
Sartana in Machete could have supplied the header quote if there wasn't already one:
"Well, there's the law and there's what's right. I'm gonna do what's right."
The Phantom Menace has Qui-Gon Jinn, who will defy the council to train Anakin because he believes the boy is the "Chosen One." Never mind how bad that went. He was sure he was doing the right thing at the time.
Heimdall in Thor is ordered by the temporary king Loki to not open the Bifrost to anyone. When the Warriors Three and Sif decide to break the rules and go anyways and tell this to Heimdall, the latter simply replies with a "Good!" and walks away. After readying the Bifrost, of course.
Nick Fury does this in The Avengers when the WSC orders a nuclear strike on Manhattan to stop the Chitauri invasion.
Nick Fury: I recognize that the council has made a decision. But given that it's a stupid-ass decision, I have elected to ignore it.
The Dark Knight Rises: Officer John Blake often goes against his superiors because of them making questionable decisions in handling Bane's attacks on Gotham. He rescues Commissioner Gordon and gets inducted into the Major Crimes Unit, and his decision to help Gotham no matter what others say comes to a head when it becomes the target of an imminent nuclear explosion.
Deconstructed in SWAT, where it's shown what happens when a stunt like this doesn't go as planned. In the opening scene of the movie, the audience sees a bank robbery that quickly turns into a tense hostage situation, and they're introduced to respected SWAT officer Brian Gamble, who disobeys a "hold" order to save one of the hostages from certain death while his superiors insist on negotiating with the robbers. It's the kind of stunt that practically all action movies depict heroically — except Gamble screws up and accidentally shoots the hostage, causing a PR nightmare for the police department, destroying Gamble's relationship with his partner, and leading to his expulsion from the force, whereupon he goes rogue and becomes a bitter career criminal. Turns out Gamble's actually the villain of the movie. Oops.
Bad Boys 2: Marcus's sister is kidnapped by the drug lord and taken to Cuba. Mike and Marcus intend to go rescue her themselves, but are then aided by a loose-knit group of law enforcement agents who decide to risk violating international law.
DEA Agent Snell: We don't know you, but you look like you're about to do something stupid. I'm in.
The White Rose activists in the fact-based Sophie Scholl: The Final Days, which details the arrest and interrogation of a group of German anti-fascists in 1943. The titular Sophie basically quotes this trope, albeit more eloquently, when her interrogator admonishes her for breaking the law.
Upldr has the main character going rogue to break all of the ethical rules to finish his project. In his mind, he thought he was doing the right thing.
RIPD: Despite being suspended for the fiasco with Fat Elvis, Nick and Roy continue working their leads on the gold.
Alien Nation: George Francisco is a by-the-book Newcomer police detective...until he finds out some of his race are manufacturing a narcotic called jabroka. Then he sets out to stop them by any means necessary; his philosophy becomes "Fuck procedure."