Katara being turned into arm candy or a prize for Aang has not gone over well with her fans.
Aang spends a lot of time with the Official Avatar Aang Fan Club and ignores Katara in the process. She wants attention from her boyfriend because he is constantly hanging around other girls with crushes on him and who often insult her. She becomes upset at this but instead of telling Aang about it, she just internalises her feelings and learns that she is in the wrong for being jealous largely due to Chickification. There are multiple panels where Katara is just left to the side being sad while Aang is left completely oblivious and having the time of his life with his fangirls.
In their final conversation, Aang is "handling" Zuko. He takes an ill and possibly suicidal young man, transports him to the tea shop that hosts the few good memories in his troubled life, lets him see his beloved uncle and sip said uncle's tea, and flatters Zuko with praise about how Zuko was totally in the right about what they'd been fighting over. Aang's statements, however, concede nothing substantial to Zuko. The Fire Nation's single desire was to retain its colonies, and Aang flatly states that no such thing will happen, and that instead the issue will be handed off to a multiparty summit. Whatever the future outcome, Aang walks away from the meeting with three victories to Zuko's zero: he avoids a war, he gets Zuko to implicitly forsake any future claim to sole sovereignty over the colonies, and, more importantly, he gets Zuko back onto his side as a friend. Zuko could arguably claim to have secured the continued residence of the colonists in their homes, but their way of life is going to change with the Fire Nation no longer in charge.
Was Suki's concern over Zuko because of a Bodyguard Crush? Or because of concern for a (platonic) friend she grew close with during their travels in the Gaang?
Broken Base: Just exactly how canon the books are is a debated topic in the fandom; some fans even feel these books almost as a fanfiction, and bringing it up risks starting a fan war. (Though not nearly on the same scale as the ship wars...)
As of the third trilogy Fan Hate has died down and the comics have become more accepted into canon. Still there is discourse on whether they are a good addition or not. With the end of The Legend of Korra, excitement for the comics has risen as they have become the sole source of continuity for the Franchise, with many fans expressing a desire to see a Korra comic.
The entire Katara/Aang romance is this (then again, not a lot of fans liked it in the original series either). Is it cute or downright annoying with the non-stop "sweetie" pet names and oogieness?
Creator's Pet: Gene Yang has described Sneers as his favorite Freedom Fighter, which is why he brought him back for the trilogy. Yang also, incidentally, paired Sneers with a hot girlfriend (Kori) and had multiple characters lampshade how awesome Sneers was for landing a girlfriend seemingly out of his league.
The Scrappy: Kori gets this from fans for her insistance on solving her problems with violence and assassination attempts on Zuko's life. That and the fact that she's a Karma Houdini. She tried to kill the freaking Fire Lord and yet she's free to organize a resistance in the colony.
Shipping Bed Death: Many people feel Aang and Katara have fallen into this, including Sokka and Toph. Once again, "oogie".
Some Anvils Need to Be Dropped: Aang's initial rejection of the Air Nomad fan club and eventual acceptance and decision to educate them and set them on a better path as a commentary on cultural appropriation. Cultural appropriation can be deeply insulting, but the possibility of good intentions needs to be considered.
Also the main story of Fire Nation Colonies and Harmony Restoration Movement, which the former had to be forcefully moved despite having assimilated with Earth Kingdom culture, showed the difficulty in reparation to Earth Kingdom for previous atrocities along with Fire Nations citizens who were decent and had Gone Native. Considering that many countries during the Post-Colonialism had controversial removal of non-nationals along with racial tensions in Real Life, this is something to be learned for many readers on the complications in nations that had gained independence from colonists.
Tastes Like Diabetes: Aang calling Katara "sweetie" in several panels. Ick. This is even lampshaded by Sokka, who calls it "Oogie".
Zuko, when will you learn? You ask someone like Ozai for advice on how to rule a kingdom? Why not ask your loyal and not psychotic uncle instead? Answer: Because for all of his hard-earned intelligence and maturity, Zuko "...is still a stubborn boy." Good thing Iroh's fully aware of this.