"Odd. I didn't think we sporked stories from Fiction Press. I mean, this must be from Fiction Press if it's an original story, and it must be an original story, because it has nothing in common with any existing fandom I've heard or seen. Weird coincidence how two of the characters have names from the Harry Potter series."
One example is the Lust Over Pendle series of Harry Potter fanfiction by A.J. Hall, a series of suspense/mystery novels using Draco and Neville as a couple solving mysteries, effectively turning them into the Hardy Boys. Magic is hardly mentioned at all, everyone constantly interacts with Muggles and one climax features a "magical" nuclear device that doesn't seem very magical at all. In other words, if you changed the name and dropped what little premise of magic actually included in the story, you'd have a pretty good original novel. With the premise sticking halfway to it, readers keep on forgetting that this is supposed to be a Harry Potter fanfiction even with the names used.
In one fic, Harry wasn't just extremely OOC, but also got a complete makeover and a new name. All canon magic was soon forgotten in favour of spells invented by the author (which didn't really fit into the canon world). The only other canon character in that fic was Draco Malfoy, who had also been given a new name and appearance.
And then there's My Immortal, which actually does replace the wizarding subjects with Muggle subjects as well as change most of the characters' names. And as for the characters who get to keep their canon names, they're still almost never spelled correctly.
There's a fic called Swan Princess. It's a word-for-word transcript of the film The Swan Princess with the names changed. Harry is in the place of Prince Derek and Draco is in the place of Princess Odette, meaning the story takes place in a world where marriages are arranged for same-sex children with the apparent assumption both will be gay when they grow up. (No matter that half the point of an arranged marriage is to produce children, not that the obvious obstacle there is necessarily an issue in fanfic.) Lucius Malfoy and Lily Potter are completely out of character, acting chummy with their lines stolen from King William and Queen Uberta. Despite still being male, Draco is referred to as a "princess" ("Then, happily a son was born... a princess. And he was given the name Draco." [sic])
A vast, vast majority of Gundam Wing fanfiction consists of taking the Gundam Pilots and plopping them into a completely different setting just to facilitate them getting hot and heavy with each other. The number of fics actually dealing with the plot and events of the series, let alone set in the After Colony universe, is a definite minority. Die for Our Ship and Wimpification take care of most of the rest of the canon.
Pretty much any fic with Fire Emblem's Marth turns him into a Possession Sue due to the Fangirl author not playing Shadow Dragon or even doing any research. Roy and Ike suffer from this as well, but it's not as bad.
The Simpsons has plenty of Transplanted Character Fic. Examples include Our Stay With Sideshow (which made the Simpsons kids WW2 evacuees), Maid in Springfield (which aged some of the kid characters up and made them the staff of a New York hotel) andDystopia (which was set in a Bad Future).
Family Guy also has several, generally with American Dad! and The Cleveland Show mixed in too. One notorious one removed the character's familial ties, got rid of Brian and Stewie and made the characters high school students.
This trope is particularly popular in the My Little Pony: Friendship Is Magic fandom, and comes in several flavors to boot. Even in the Fandom's earliest days it was popular to plop the cast into strange settings, like Earth. Then Human in Equestria fics started appearing and were for a brief time loved by the fandom, until they became excuses for Mary Sue-esque self inserts. To Rectify this issue, many authors starting putting in various Crossover characters and keeping with the Human in Equestria theme. In a strange version of this Trope, the main cast might not play an incredibly important role if they play one at all, with our transplanted Heros only dealing with living in a world of horses. Instead the Crossover Character will bunk down with an OC or a background Pony, so an MLP fic without any major MLP characters. Today most readers don't read Human in Equestria fics unless they star a Video Game character, a War Hero, or an Astronaut as the human (because Astronauts and War Heros are Cool).
Many BIONICLE comedy fanfics feature the characters going to high school or opening a resturant/hotel together.
A lot of Redwall fic is original furry fiction put on the fanfic sites to get more reviews. Soulless Shell is a prime example, with the only connection to the Redwall universe being that rats and the other vermin are the evil races. Main character Leif has the ability to shoot laser beams, teleport and turn into a demon, while no magical powers exist in the Redwall universe.
There's one Star Wars fanfic that involves a girl from Earth (in which Star Wars does exist) finding out that she's actually a Jedi after her parents die in a car crash, so she's taken to Coruscant to be trained as a Jedi. She befriends Anakin Skywalker and Obi-Wan Kenobi, and later takes them to Earth where Anakin finds out that he will become Darth Vader by looking at the box for the Revenge of the Sith Xbox game. The author later revealed that it was actually an original story that she was writing and attempting to get published about a school for magic.
A disturbing number of Pokémon shipping fics include absolutely no Pokemon, even in passing mention, background, or setting. Note that this completely undermines the entire point and title of the franchise. A similar problem is omnipresent to a certain degree in Digimon, Yu-Gi-Oh!, and Bakugan.
Popular series Kingdom Hearts gets this very often. Most of the stories on Fanfiction.net's front page for it will usually be nothing about the game at all, with practically every character being in an Alternate Universe setting, acting little to nothing like themselves...and gay.
In Axis Powers Hetalia fandom, the fics that fall into this trope the most frequently are...not the human AU ones, the majority of which actually at least try to keep the characters IC and their relationships reminiscent of their canon ones, even in High School AU (which was actually originated by the author himself!). No, the fics that most often invoke this trope are the ones that do keep the "characters are Anthropomorphic Personifications of nations" premise...but completely rewrite the characters' personalities (often for the worse) for the sake of "historical accuracy" (i.e. "fitting the fanwriters' own ideas about history, even if they directly contradict established APH canon" or even "using 'history' as an excuse to make a character Die for Our Ship or become Ron the Death Eater"), to the point where the characters are essentially national Anthropomorphic Personifications that were created by someone other than the series' author.
Keep in mind also that Tropes Are Not Bad as well in that there are also a good deal of fics out there (whether the human or personified AU kind) which manage to pull this off fairly well at least. Grey Skies Over London for instance keeps America and England's personalities intact while building over them as the Alternate History surrounding them changes even further.
There are several AU Toy Story fics that depict the toys as humans. Quite a few of them feature these characters as high school students. There are many AU fics which depict non-human characters as humans. Most of these depict those characters as high school students.
While The Return is very well written and interesting, it has almost nothing whatsoever to do with its ostensible sources, Ranma ˝ and Sailor Moon. The Ranma elements suffer from changed setting and skillset, and all the characters' personalities being utterly destroyed and transformed into something otherin-story, while the Sailor Moon characters come out with personalities and abilities more or less intact, but story tone so dark as to make it almost unrecognizable.
Similarly, On A Clear Day You Can See Forever is a great story, but Ranma's been drastically changed by having almost literally been through hell, and the 'parallel' universe that the Sailor Moon characters are from is a dark Urban Fantasy complete with gods, mages, vampires, werewolves, and airships, with only lingering archetypal resemblances to the source material.
Final Fantasy VIIIThe Altimate Rewrite is not so much an attempt at passing an original story off as a fanfic as it is the writer's attempt to improve on the original by making extensive changes to just about every element of the game's setting, characters, and plot. The result is, in any case, very far removed from its source material.
The Homestuck fandom has spun off all sorts of bizarre A Us with little resemblance to the original other than the names and descriptions of characters. High school, noir, asylum, cop drama, you name it. A particularly bizarre example would be Homestuck High, which turns rapidly from a High School AU into a mishmash of battling demons and terrible porn.
One popular fanfic pitted John as a Humongous Mecha pilot against the rude alien child soldier Karkat in a setting inspired by Neon Genesis Evangelion; another popular AU used by many different authors makes the trolls into a cute pet race owned by the humans.
The entire 'uberfic' movement within the Xena fandom quite often (but not always) even did away with the names of the canon characters, and simply wrote original fiction with characters who archetypally resembled Xena and Gabrielle. Many of these stories did go on to be published as small-press fiction. If you encounter a lesbian romance that features a strong dark-haired woman with a painful past and an innocent blonde who believes in helping others and has had an unsatisfying relationship with an ex-or-dead husband, you've probably found an Uber.
The Pokémon fanfic author StaraptorEmpoleon is infamous for taking this trope to its logical extreme: it's not uncommon for her to change everything about the Pokémon characters often up to and including their names such that it often seems like the only thing they have that's connected to their canon selves is their star pokemon (if that)!
Overwhelmingly common in Cardcaptor Sakura fanfiction. Many fic summaries boil down to "Sakura is X, Syaoran is Y, Z happens, will love result?". Several authors have admitted to not reading or watching more than an episode of the canon, basing their fics primarily on other fics.
Quite a few The Lion King "crossovers" are like this. The author simply takes the characters from another show or movie and have them run through the Lion King plot. In fact, many (if not all) Disney fanmake fics are like this.
A Happy Dream zigzags this. The core premise of the original series (magical girls and witches) aren't real... but they are the subject of a ton of Mythology Gags, culminating in a circus resembling Walpurgisnacht and staffed by humanized versions of the witches. The characters also go back and forth in terms of characterization; Homura in particular seems to be something of an unusual synthesis of her two personalities in the original.
Touhou doujin. An example: a doujin that takes place in normal world, the main character is Youmu, and she is doing some errand for her landlady(?) Yuyuko. In doing so, she meets a lot of Touhou characters doing normal things that normal people do. For example, Eirin is running a drug store. The doujin would have made a very poignant philosophical discourse about the impermanent nature of everything. (In Japanese, they call this mono no aware.) It should be noted that this is almost a majority of Touhou doujinshi.
Sailor Moon fandom triggered rants about the frequency of the UsaMamo AU Romance, transplanting the characters into cookie-cutter romance novel situations.
The Star Trek: The Next Generation episode "Sub Rosa" was initially pitched by a freelance writer, later developed into a story credit by Jeri Taylor and a teleplay by Brannon Braga. Many felt the final product was basically the Anne Rice novel The Witching Hour - only with Doctor Crusher and a Techno Babble "ghost." Taylor denied Rice's novel as the inspiration, instead saying they were simply doing an homage to The Innocents movie and utilizing various gothic tropes.
Cartoon Network's [adult swim] late night "adult" programming block has a few shows wherein the entire premise is the as-blatant-as-possible application of this trope to the original source material, usually utterly forgettable and short-lived Hanna-Barbera cartoon shows from the Animation Age Ghetto: