"The Ember Island Players" proves this isn't all in the fans' heads and that the writers are actually conscious of how this comes off.
In the episode where June helps Zuko find the Avatar, June asks him, "What happened? Your girlfriend run off on you?" Zuko replies, "It's not the girl I'm after; it's the bald monk she's traveling with."
Katara:(his love interest, keep in mind) No, way! I'm sure he only did it so he could capture you himself!
Sokka: Yeah, face it, Aang — you're nothing but a big prize to him.
The picture features Aang trying to take some of Zuko's flame because his went out, and getting way too close to him in order to do so.
Zuko says that Aang gives him a purpose in life and even makes his firebending more powerful. His "inner fire," as he says. Wow.
In the Season 2 premier, Aang's struggling with whether or not it's a good idea to try to defeat the Fire Lord by using the Avatar State and having nightmares about how much the power scares him. It's not until right after he has a nightmare about attacking Zuko while in the Avatar State, however, that he decides it's too dangerous and wants to stop messing with it.
Azula's not-at-all subtle tendency to surround herself with stunningly gorgeous handmaidens sporting bare midriffs.
It may be a bit of a stretch, but in the Final Battle and even some of their earlier fights, Azula's behavior towards Katara can be interpreted as having some... almost jealous or possessive overtones. Her "Filthy peasant" is usually said in such a way that it almost qualifies as Terms of Endangerment, and her tendency to go out of her way to hurt Katara and/or gloat over her pain could indicate a borderline obsession.
Waterbender + Earthbender = only a matter of time before they got to mudwrestling each other.
Azula's downright infamous "favourite prisoner" remark about Suki, said in a tone of voice that makes it very difficult to interpret as anything other than this. Granted, it's Azula, so she could easily be lying to make Sokka angry, but this isn't the only thing hinting at Azula being a Depraved Bisexual.
And even if she was lying, that means the implication was likely intentional either way.
Ty Lee mentioning that she bonded with the Kyoshi Warriors in prison sounds like a lesbians in prison movie, especially since she then grabs two of them for a hug and they all get humongous grins on their faces. 'Cept only Ty Lee's grin is unabashedly happy; the other two look worried, as if they're wondering if she'll let slip what happened in there... "We must never tell a soul."
Roku and Sozin. When Sozin finds out that Roku is leaving to become an Avatar, Sozin privately gives Roku his crown as a parting gift, hoping that Roku will at least be able to have that as a possession (because Avatars are supposed to not need material goods). "See that little crown Roku wears? That means he's the king of queen Sozin's world." Sozin "borrows" Roku during the latter's freaking wedding (and his wife does not seem too pleased about it). While they're sharing some alone time, Sozin starts trying to convince Roku that he's been doing some thinking, and wants Roku to stay by his side and rule the world with him. Yes, he decides to do this the exact same day Roku is getting married, taking him away from his wife to tell him this privately, which... has some rather interesting implications. And then there's how they hang out and spar with each shirtless, with Sozin having the habit of saving Roku from falling by pulling him to him. There's also how Sozin kept trying to get Roku to join him in taking over, only to get very angry after being refused so many times... along with how Sozin didn't get married until much later.
Its worth pointing out that the only thing mentioned above that is Justifiedin Context is the shirtless sparring (they're firebenders, and shirts are flammable.)
It's also worth pointing out that Sozins last testament (the one Zuko reads) is all about his relationship with Roku.
As mentioned above, June does a disproportionate amount of smirking over Katara. And then there's her reaction to meeting her:
"No wonder she left. She's way too pretty for you."
Heterosexual Life-Partners Azula and Ty Lee act as a foil to Zuko and Mai in Lower Deck Episodes starring the villains, occasionally with quasi-romantic lines like Ty Lee reassuringly calling Azula "the smartest, prettiest, most perfect girl in the world". During a scene in "The Beach", Azula shows a warmth to Ty Lee that she never shows to anyone else in the series. Of course, none of this stops Ty Lee from betraying Azula for Mai's sake.
On that note, it was Ty Lee's betrayal that was the final trigger for the beginning of Azula's Villainous Breakdown. Mai's betrayal she was angry about, but that was because it went against everything she personally believed rather than any feelings for Mai. With Ty Lee, on the other hand... Azula is first outright horrified then flat-out livid about that betrayal. It could be because she thought Ty Lee was far less likely to turn on her than Mai, but if one chooses Alternate Character Interpretation....
It's clear from the girls' expressions and the way the dialogue was delivered that Azula was about to kill Mai before Ty Lee intervened — it was only after Ty Lee joined Mai that she decided to be inexplicably merciful and have them imprisoned instead.
The time Toph mistook Suki for Sokka and kissed her on the cheek. Suki didn't seem to mind.
Sokka and Zuko don't interact much, but when they do... well, let's just say Boiling Rock is also known as Boiling Cock.
The famous moment in "The Southern Raiders" when Zuko enters Sokka's tent to find Sokka lying on his stomach with a rose in his mouth. Sokka greets him by saying "Well, heeeeelllllooooo..." Sure, it was for Suki...
Hakoda and Bato, to the point where some fans call them "Katara and Sokka's two daddies". The ship-happy fandom was pleased to hear in "Bato of the Water Tribe" that Hakoda carried Bato to the abbey when he was wounded, and then there are misleading screen shots like this one◊ where it looks like they're nuzzling foreheads.
Ty Lee has a persistent habit of glomping Mai, which annoys the latter to no end.
Rather far fetched, perhaps, but it turns out that the Ocean Spirit is female, that is, embodies the Yin side of the relationship between the Moon and the Ocean. The Moon Spirit had been male until Zhao killed him. So now Yue and La are intertwined in eternal dance with romantic connotations around each other.