Mai: Good Lord, her treatment by most of the A:TLA Fan Dumb defines this trope. At first, most Zuko/Katara shippers were apathetic to little!Mai. Even when she came back, all grown up, in season 2 to help Azula pursue the Gaang, most Zutarians viewed her as just a slightly annoying side-character... until season 3 episode 1, when the audience first saw her make out with Zuko. Then Mai became the target of the great Zutarian Fan Dumb. As Mai and Zuko's canon relationship blossomed, the Zutarian bashing only intensified. Katara was the character these fangirls identified with, and Zuko was the character these fangirls wanted to tap. When Mai moved in on the boy they wanted, the Zutarians hit her with everything they had, as if this had happened in real life. There's even fanart that lampshades the Fan Dumb about Mai.
Since they can't kill off the Kid Hero, many Zutara shippers make Aang so childish that he makes Honey-sempai look like a man of deep gravitas. The most rabid of these shippers go so far as to call him "Bryke's special little Gary Stu" or "self-insert golden boy" for daring to get his happy ending with the girl he likes; deviantART and Livejournal are littered with angry rants and "intelligent" essays blasting the Aang/Katara pairing, taking Alternative Character Interpretation to extremes by twisting Aang into a selfish little bastard who "only won Katara as a prize for being the Special Hero", accusing Kataang of being "creepy" or "sexist" and clinging to the rejected kiss scene in "The Ember Island Players" as "evidence" of this, completely discounting two and a half seasons of friendship and development.
Others assume Avatars must either be Celibate Heroes or else rapidly became villainous. Until Word of God and a canon episode discredited this idea implicitly, this was one of the primary means of pushing Aang to the side; due to ancillary material, it was fiercely contested by Kataang shippers during the long drought of new TV eps of 2007. Of course, the "celibate" angle has now become Hilarious in Hindsight, as the live-action The Last Airbender film actually does portray the idea that Avatars can never have families as a plot-point (namely, it was why Aang fled his responsibility as the Avatar in the film, as opposed to being afraid his life would change for the worse, and that he would be taken away from his mentor/father figure, Gyatso). Certain fans have noted this, and have used it to try and support this angle (despite it being canon to the film, and only the film), but on the other hand, most fans would rather ignore the film entirely.
Other shippers discounted Aang because he was 12 years old. They usually assigned a big sister/little brother type relationship to Aang and Katara, villainizing neither and viewing their relationship as one that they would naturally grow out of as they got older, or claiming that a romance between them would be creepy Shotacon even though Katara was only two years older(forgetting that Zuko is two years older than Katara) than Aang. There's also the camp that claims that because Katara has a maternal personality towards the team, Aang's attraction to her is "Oedipal"...which makes no sense since Katara is not actually his mother (biologically or otherwise). Then the sequel series had Word of God identifying Tenzin as Aang and Katara's son before it aired, thus confirming that their relationship did last beyond the "teenage hormones" stage. Needless to say, this simple declaration set the Kataang vs. Zutara wars aflame again.
A few Toph/Katara shippers are nasty to Aang in their fanfics, twisting him into a jealous asshole who just wants to hurt Toph for "stealing" Katara from him.
If Aang isn't rendered childish, expect to see him sneered at for being a self-righteous prick who doesn't understand Katara's struggles with revenge and bloodlust in the episode "The Southern Raiders," even though Aang explicitly points out that he understands, and that he's advising Katara as any friend would against a path that is dangerous and immoral.
Somewhat related to the forgiveness issue, Aang is sometimes portrayed as trying to force his Air Nomad traditions onto others, even when they disagree (primarily Katara). His decision not to kill Ozai (which, admittedly, is controversial for other reasons) is portrayed as being solely motivated by those traditions, and an irresponsible act at that.
Jin: Jin had quite a few attacks on her from a vocal section of the Zutara fans and even some early Maiko fans. Most of it called her a slut or a gold digger (even though she canonically didn't know Zuko was a prince), apparently because she is more endowed than most of the girls. Rage against her was relatively short lived — it quickly became clear that she would not be a recurring character or have any lasting romantic interest in Zuko.
Zuko: A section of the Katara/Aang fandom have become openly hostile to Zuko due to the extreme popularity of the non-canon Zuko/Katara ship, calling him emo and ugly. The Sokka/Suki shippers, seem to be readying their guns to do this too, as there have been Ship Teases of a Zuko/Suki relationship in the Promise comics.
Katara: Extreme Maiko and Aang/Toph fans bash Katara as a Canon Sue who became far too good at water bending far too quickly.
Suki: You'd be surprised at the number of Toph/Sokka shippers who were saddened and angered to discover that Suki was still alive. Now they're the poster Love Triangle for Avril Lavigne's "Girlfriend." A few Sokka/Azula shippers have recently come out of the woodwork to bash Suki as well.
Song: Song got similar treatment as Jin by diehard Zutara shippers and a few Maiko shippers back in the day. The most common accusation was that she was a Mary Sue; nowadays, Song is usually just flat-out ignored since she only showed up in one episode and never appeared or got referenced again.
This girl may never get a reprieve. Even after proving her loyalty to the Krew and her personal ideals by going against her own beloved father, some of the crazier sections of the Makorra fanbase complain about how "slutty" she is and even through the Book 1 finale kept insisting she would turn out to be an Equalist spy. Even though Asami has proven herself a good ally to Korra time and time again by fighting against her father and even though Asami stuck by her convictions despite Mako and Korra's generally bad treatment of her, there are still people who insist that Asami will "turn evil" in Book 2 because they're still upset that she anchored Makorra for a good portion of Book 1.
The Asami hate came back in full force for Book 2, when Mako and Asami got back together again. People calling her a slut and all sorts of terms claiming she got in the way of Makorra, despite the fact that she and Mako got back together AFTER he broke up with Korra. (Though it didn't help that Mako hooked up with Asami only a week after he broke up with Korra, something Bolin calls him out on.)
This almost reaches the levels of Fan Dumb because Mako and Korra got together in a lot shorter amount of time after Asami and Mako's sorta break up.
Pema: Despite the fact that we don’t know all the details of Lin and Tenzin’s breakup, the word “homewrecker” still gets tossed around by some of the Linzin fans. Especially unfortunate is the desperate claim that Tenzin is now obligated to leave his wife of many years and three children (or better said, four children) to get back with his old girlfriend Lin, because it makes Lin's worth as a whole hang less on her Lady of War side and more on whether she gets Tenzin or not.
Mako: There are many perfectly legitimate reasons to dislike Mako or his portrayal in the series. Hating him just because Korra happens to like him (and wishing he would die in a fire) is not one of those reasons. But don’t let certain vocal sections of the Borra, Tahnorra, and Korrasami fanbases tell you that.
Makorra shippers, now Armed with Canon, declared war on anyone that could be potentially seen as an obstacle to their ship even though they got what they wanted anyway.
Book 2's ending then rolled out some new canon for the opposition.
Mako almost instantly elevated to Ron the Death Eater status over the hook-up, as he was accused of being a jerk who two-times Asami and when taken to even further extremes, portrayed as an abusive monster. He's also been accused of being a Mary Sue or a Creator's Pet simply because...he hooked up with Korra after breaking up with Asami. Not shockingly, some of the most vocal of these were Borra and Korrasami shippers, though really, anyone that wasn't shipping Makorra went into shipping seizures. Not helping was that Mako and Asami's breakup was done in a very "blink and you'll miss it" fashion, meaning that it can easily come off like Mako didn't bother to break up with her before making a move on Korra.