Subverted at least once in an old cartoon - Bluto enters Olive's room, and a scuffle breaks out, Olive is crying for help - when Popeye enters, Olive is still yelling while clubbing an unconscious Bluto with a skillet.
Buddy saves Cookie from a Bluto-like character...
Mighty Mouse saves Pearl Pureheart from Oil Can Harry...
Bimbo or Koko saves Betty Boop from various baddies...
Bosko saves Honey from more various baddies...
The pre-Mighty Mouse Terrytoons mouse lead saves his girlfriend from more various baddies...
Toby the Pup saves Tessie from more various baddies...
Flip the Frog saves Flap, Kitty, and Fifi from even more various baddies...
Julius saves Alice from Pegleg Pete...
Oswald saves Sadie from Pegleg Pete....
Mickey Mouse saves Minnie from Pegleg Pete (Disney only had one recurring villain... pass it on).
Subverted in Pioneer Days and Building a Building, where Mickey tries to rescue Minnie but is captured himself, whereupon Minnie breaks free on her own and rescues Mickey.
Even his dog, Pluto, gets in on the trope, saving love interests Dinah and Fifi a few odd times.
Subverted along with everything else in Chuck Jones' melodrama parody The Dover Boys as their fiancée, Dora Standpipe, is abducted by villain Dan Backslide - she doesn't break the pace of her cries for help even as she demolishes Dan.
Subverted in the movie Batman & Mr. Freeze - Subzero. Yes, Plucky Girl Barbara a.k.a. Batgirl gets kidnapped, but she's so competent that she kicks the asses of her captors (Mr. Freeze being one of them) multiple times, and would have escaped on her own just fine if it wasn't for the fact that she was in the middle of the friggin' ocean. In fact, she is probably more useful in the movie than even Batman and Robin.
The entire episode of "Beauty Marked" in Danny Phantom was made in order to subvert this as much as possible. While Danny and Tucker are under the mindset that the kidnapped Sam needs rescuing, she managed to figure a way out just fine. It is their meddling that gets her captured again/still.
Gosalyn Waddlemeyer-Mallard in the first pilot, and of course, some certain occasions from the Classic Disney 90s animated series Darkwing Duck.
Xandir P Whifflebottom in Drawn Together is a video game character "on a neverending quest to save my girlfriend". When she discovers he's gay she refuses to be rescued by him. Shortly afterwards his boyfriend is kidnapped and he remarks "Dare I say it? I, Xandir, am on a never ending quest to save my boyfriend!"
Nell Fenwick on Dudley Do-Right is a parodic composite of the woman tied to train tracks in the gothic genre.
Subverted in the Foster's Home for Imaginary Friends special, "Destination Imagination"; the plot was to save Frankie after she was kidnapped by an imaginary friend who controls a trippy world inside a toy box, but near the end it's revealed that she wasn't kidnapped at all, and that she willingly stayed with the imaginary friend to keep him company. But at the climax, when the imaginary friend has a Villainous Breakdown and becomes a monster (thanks to a verbal lashing by Mr. Herriman), the characters fight him to protect Frankie from being trapped in the imaginary world forever. However, the gang are roundly defeated by the monster, and ultimately it's Frankie who becomes the hero of the story: not only does she distract the monster so that the gang can escape the toybox — she briefly stays behind but soon escapes on her own, thus completing the subversion — but she chooses to free the lonely friend as well, having offered to bring him to Foster's, where he can have all the friends in the world.
The Herculoids. Tara. When she wasn't getting threatened or captured by the Monster of the Week, she was being kidnapped by the lizard-snake people to be their brainwashed queen. Mitigated by her usually trying to run from danger when she isn't being ambushed. Zandor and the Herculoids always managed to rescue her -and Tara's so nice and generally sensible it's understandable why they keep doing it.
Kim Possible: Every main character (and some of the villains) have been in this situation. Kim. Ron. The Cheerleaders. Bonnie. Kim's Dad. Kim's Grandmother. The Tweebs. Ron's Dad. Shego. Drakken. Shego's little brothers. It's a requirement of this show that you get captured at least once.
Isabella finds herself in this role to Phineas a handful of times (and vice versa), part of the constant ship teasing between the two over the course of the show's run. Examples include playing the superhero to her Lois Lane-style Intrepid Reporter in "The Beak", rescuing her from a mechanical bug in "Hide and Seek", and building a haunted house to break her "curse" of hiccups in "One Good Sacre Ought to Do It".
Two of Isabella's Elseworld equivalents also fall into this category. Princess Isabella of feudal China is abducted by Doofus Khan, and Phineas and Ferb learn the Way of the Platypus to mount a rescue mission. Lampshaded In 1914 Panama, where she literally carries around business cards designating herself as a fully certified damsel, as well as an Ingenue, Femme Fatale, and tile and grout installer. Later on, she hands them another: "Dirty Double Crosser", as it's revealed she's in league with the Doofenshmirtz counterpart to keep him from hurting her mother.
There were a few episodes of The Powerpuff Girls in which the girls themselves needed to be rescued (e.g., "Buttercrush", "Twisted Sister", "Stray Bullet").
Subverted on ReBoot. One game sets up this plot with Bob as the rescuing knight and implies Dot is the distressed damsel. The subversion is that Enzo is the distressed damsel and Dot is another rescuing knight.
Enzo: I don't want to be a damsel in this dress!
Ashi from Samurai Jack Season 5 is an interesting case; normally a straight brawler, she discovers in the penultimate episode that she is Aku's biological daughter. Aku takes full advantage of this, mutating her into a demon while her good self remains trapped and fully aware for Jack to try and save.
Shannon:[referring to the Blake family] Ours is a rich history of falling through trap doors, finding dangerous secret passageways, and getting caught in overly complicated booby traps of our own design. The one thing we Blakes are known for is being... how should I put this... danger-prone. Daphne:[everyone glances at her] What?
In a episode of Sonic Boom called "Fortress of Squallitude", Amy is captured by Eggman and Amy is forced to make Eggman's fortress beautiful. Sonic and co. have to save Amy.
In the episode "Sleeping Giant", after Rocky takes a liking to Sticks' singing, Sticks is captured by Rocky and she is forced to sing for Rocky. Sonic and co. have to save Sticks.
In the episode "Closed Door Policy", Sticks is captured by the frog warriors, leaving Sonic and co. to rescue Sticks.
Talking of the Sonic series, Sonic Sat AM, Sally also becomes this in a few episodes, most notable in "Sonic and Sally" where she had a robot take her place and where she was almost turned into a robot herself, but was rescued by Sonic Just in Time. The second season tried harder to make her Sonic's equal so this didn't happen.
Eventually lampshaded by Shredder when Kraig suggests to Shredder to send the stone warriors Shredder made to grab April O'Neil. Shredder's response is this:
Shredder: Hmm, not a bad idea. Although we've done that at least thirty times before, but hey who's counting?
Also lampshaded in the Turtles Forever crossover movie. The 1980s turtles stop to save April in their home dimension and explain that they save April at least once a day.
In the Nick series April starts out as this, but as her fighting skills have been improving she's growing out of it.
In the shows Tom & Jerry Kids and Droopy Master Detective, Miss Vavoom is always getting kidnapped by McWolf or any other villain who lusts after her.
Elita One gets this with her love interest Optimus Prime in Transformers: Generation 1. To lure Optimus to his doom, the Decepticons capture Elita. However, when Optimus arrives, he gets captured himself and Elita first has to save him before he can manage to save her. It's pretty 50/50 with them.