"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.
Jet and Zuko have their own amount of Foe Yay. It started out as Ho Yay, with Jet inviting a boy he knew nothing about to come "liberate" some food for the rest of the refugees, later inviting him to join his group... right up until Jet became convinced Zuko and his uncle were firebenders. Then it devolved into what could be easily interpreted as Stalker with a Crush behavior. Interestingly, during Jet's full blown meltdown under Lake Logai, one of the many things he thinks about while coming out of it is Zuko.
Waterbender + Earthbender = only a matter of time before they got to mudwrestling each other ("The Runaway").
Azula generates subtext with everyone, regardless of gender.
Mai and Ty Lee were Azula's only friends, she crossed a continent to recruit them specifically, and trusts them enough to be relatively open about her concerns and plans around them. It's Azula so it's hard to know, but her Villainous Breakdown being triggered by their betrayal and a line in the Grand Finale shows that she really did care about them.
"You'll all leave, just like Mai and Ty Lee did..."
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. However, none of this stops Ty Lee from betraying Azula for Mai's sake, which was the last straw to trigger the beginning of Azula's Villainous Breakdown. Mai surprised her, but with Ty Lee Azula is first outright horrified then flat-out livid.
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.
Her infamous remark that Suki is her "favourite prisoner" (when the last time we saw Suki the two were fighting, before Azula took her clothes), said in a tone of voice that makes it very difficult to interpret as anything other than this. She could easily be lying to make Sokka angry, but that means the implication was likely intentional either way.
Azula in general has a not-at-all subtle tendency to surround herself with stunningly gorgeous handmaidens sporting bare midriffs.
Worth noting is that Azula was originally meant to be a boy, so it's entirely possible that Ty Lee might have been intended as a "straight" love interest for the original male version of Azula. Whether that makes their dysfunctional bond more creepy or less creepy is up to the viewer.
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). 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. They hang out and spar with each shirtless (Justifiedin Context, as they're firebenders and shirts are flammable), with Sozin having the habit of saving Roku from falling by pulling him to him. 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.
Sozin's last testament (the one Zuko reads) is all about his relationship with Roku.
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."
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. Which they managed to lampshade in the episode. Chit Sang said something to them along the lines of "If you two are done cuddling..."
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.