In Disney's Peter Pan, Tinker Bell gives Captain Hook the location of the Lost Boys' home if Hook agrees to take Wendy away, and adds that he not harm Peter, to which Hook confirms that he will not "lay a finger or a hook on Peter Pan." Later, after capturing the kids minus Peter, Hook lowers a timebomb into the lair.
Hook abuses another loophole in the sequel. Since Wendy's daughter Jane is dissatisfied with her stay in Never Land, Hook gives her a whistle she can use to signal Hook and his men to set off a trap for Peter Pan at the opportune moment. Jane only agrees as long as Hook promises never to harm "a hair on Peter's head", only capturing him instead of harming. When the trap is set off (by one of the Lost Boys who finds her whistle and blows it, not knowing any better), Hook gleefully gloats over what he's going to do to Peter Pan. Jane angrily reminds him that he promised never to harm a hair on Peter's head. To keep his word, Hook plucks a single hair from Peter and gives it to Jane, saying this is the hair he said he wouldn't harm!
In Disney's The Sword in the Stone, Madam Mim immediately sets ground rules for her Shape Shifter Showdown with Merlin, among which is this: "No make-believe things", specifically, "Pink dragons and stuff." Being a foul cheat, she never had any intention of following them including at the climax when she becomes a purple dragon. When Merlin tries to call her out on this, she simply responds with, "Did I say 'No purple dragons?' Did I?!" The "and stuff..." part of her "rules" clearly implies she did, but she's beyond caring at this point.
She has no loophole to excuse her turning invisible, though.
Merlin one-ups her on this magnificently - while there's a rule that says "no turning into plants or minerals" he turns himself into a germ - which is neither - to infect her and win the duel.
Invoked by Kuzco in The Emperor's New Groove, after he has been turned into a llama and goes back on a deal sealed with a handshake: "The funny thing about shaking hands.. you need hands!" (Cheerfully waves his hooves.)
Subverted in Ratatouille: There Ain't No Rule saying a rat can't become a chef (in fact, the phrase "Anyone Can Cook" practically qualifies as Arc Words), but there is a rule that a rat can't be in a restaurant kitchen. Remy spends most of the film trying to be a chef without getting caught breaking the second rule.
Disney's The Princess and the Frog has a plot filled with excessive Loophole Abuse. First of all, there's Dr. Facilier who promises his clients something they want figuratively and then screws them over by giving it to them literally. For example, Naveen needed money, Facilier promised him green, and Naveen ended up being transformed into a frog.
Naveen can lift his curse by kissing a princess. He kisses the heroine Tiana, but since she's not a real princess, she changes into a frog as well. They are then presented a loophole way out; a kiss from Tiana's friend Charlotte will transform them both back, since Charlotte is daughter of the king of the Mardi Gras Parade and therefore temporarily a princess (as long as Mardi Gras lasts).
The most uplifting / beautiful example of this trope comes in the final scenes; as Charlotte's kiss fails to transform both back, as it was past midnight and her title of Princess had already lost its legitimacy, Naveen and Tiana decide to get married nevertheless - as frogs - and then find out why Mama Odie was shipping them so desperately; as Tiana marries Naveen she becomes a princess, and as he kisses her, he thus kisses a princess and both are transformed back to humans.
Attempted by Merida in Brave. The rules say that the firstborn of each clan have the right to compete in a contest of her choosing to determine her suitor. Since Merida is the firstborn of her clan, she declares an archery contest, puts herself forward as an entrant, and wins it decisively. Unusually for this trope, Reality Ensues; nobody accepts the result, the competing clans nearly go to war over the insult, and she still has to be betrothed in the name of peace.
Because in Aladdin: The Return of Jafar he's now bound by the rules of the genie, Jafar can't kill Aladdin himself. This forces him to rely on proxies and indirect assassination attempts where he technically doesn't lay a finger on Al.
Before that, in the original film, when Aladdin is drowning, Genie shakes his head back in forth in a nod, takes it as a "yes", and rescues him.
When stuck in the Cave of Wonders, Aladdin goads the Genie into blasting him and Abu out of there. They stop at a oasis and talk about the wishes. When Genie states he already used up one, Aladdin states he didn't wish to be out of there. The Genie did that all on his own in a show of power. The Genie is forced to consent.
Genie: Boy. I feel sheepish. [turns into a sheep] Alright, you ba-aaa-d boy, but no more freebies.
Monsters University uses this in the second event of the scare games, wherein the teams had to catch a flag without being caught by the librarian. However, there is only one rule: Don't get caught. While every team moves as silently and out-of-sight as possible, Oozuma Kappa manages to win by dividing their team to intentionally cause as much noise as possible so that the librarian will not look at Sully who just sprinted towards the flag without regards for how much noise he was making or how the flag was actually captured during the ruckus. The announcers even admit that the only rule is "Don't get caught" and their win was legit, since they technically did not actually get caught until after they had won.