For a specific, if somewhat subtle example, there are Maiev and Naisha. Naisha is one of the few people Maiev was really close to, and Naisha's death caused by Illidan, immensely intensified Maiev's hatred towards him. Also, after Illidan also rendered her brother comatose, was ultimately the reason why most of her Watchers were killed, left Maiev to die, trapped in the Tomb of Sargeras, and had her imprisoned, tortured and possibly even raped, Maiev shouts "This is for Naisha!" during the fight against Illidan in The Burning Crusade.
Skyguard Khatie, a neutral gnome quest-giver of the Sha'tari Skyguard, flirts with the player at every opportunity, regardless of the player character's gender. Crosses into outright Squick if the player happens to be playing a tauren, draenei, Forsaken, etc.
Khatie's dialogue with the player changes depending on their standing with the Skyguard, going from casual flirtation with a colleague to Khatie coming off as being somewhat tongue-tied in your presence. It's actually led to her being sort of a woobie for the World of Warcraft community.
It's much the same story with the Alliance Brewfest vendor, Belbi Quickswitch. However, the fact that she is wearing goggles that make everyone look like mostly naked male gnomes might have something to do with it.
That and, you know, all the beer.
Sentinel Sweetspring, a female Night Elf in Star's Rest, will occasionally grin and "waggle her eyebrows suggestively" at the also female Sentinel Amberline.
There's an epic quest chain in the Badlands were you are told three hilarious stories by three different people. The third guy will tell you that the first two stories weren't that great because, among other flaws, they had no hot babes in them. When you're phased to his version of the story, you are surrounded by hot babes... one of whom is a male blood elf. Should you choose to take the male Blood Elf with you during the quest, he even refers to him as a "strapping Blood Elf", so it's not another "no one can tell the Blood Elf sexes apart" joke.
The jewelcrafting daily quests in both Stormwind and Orgrimmar are given by female characters. While all five possible quests are Futurama references, "A Present for Lila" makes it sound a lot like both questgivers are employing you to make them a gift that they can give to their mutual love interest, a one-eyed female pirate captain.
Anren Shadowseeker, a night elf, and Tholo Whitehoof, a tauren, first appeared in Cataclysm as unremarkable quest givers who happened to be standing next to one another just outside the starting area in Hyjal Summit. In patch 4.2, during the Molten Front dailies, they reappear to help fight the invasion from the Firelands, and every single thing (no, really, literally every bit of dialogue or quest text, with maybe one or two exceptions) either one of them says has to do with their deep friendship and devotion to one another. By the eighth or ninth time you run into them, it's difficult for anyone, regardless of personal inclinations, to not be wondering when they're just going to start making out.
Speaking of the Molten Front dailies, one quest has you running around a battlefield healing wounded druids, who will thank you when they get up. At least one of the thank you quotes has the druid coyly suggesting they'd like to find a way to repay you later on, regardless of your character's gender.
Most pairs of male characters have to work to get to Those Two Guys status. Anren and Tholo use it as starting blocks in a 100 meter dash to the ho yay finish line.
Koltira and Thassarian, come on now.
To elaborate, Thassarian was killed by Arthas and raised as a Death Knight, then later fought with Koltira's brother, but spared his life. Koltira then fought Thassarian and tried to convince him to rejoin the side of good via "I Know You're in There Somewhere" Fight but Thassarian struck down Koltira, who was raised as a Death Knight, yet they found comradeship in undeath. Later, Thassarian joins the Alliance and Koltira joins the Horde, but they maintain their friendship despite their factions. Instead of fighting over a location, they call a temporary truce between the Alliance and Horde. At the end of the quest line, Koltira is dragged off by Sylvanas in order to remove the other loyalties he holds, and Thassarian states his intention to go after the Banshee Queen herself to find out what happened to his friend. Super-Homoerotic or worlds most epic bromance? You decide!
there's also the fact that, in the death Knight starting zone, Thassarian ask you to go deliver Koltira who was made prisoner by the Scarlet Crusade. While saying that, normally, Lich kings' servants who failed in battle are left to die. He also refer to Koltira as a sort-of brother for him. Koltira isn't surprised at all by this when you find him.
In Mists of Pandaria, there are two women, Ann and Marsha Stockton, standing in the inn in the Shrine of the Seven Stars. Neither have any unique dialogue, but one's a dwarf, one's a human, they have the same last name, and they occasionally use the /flirt emote at one another. They're widely theorized to be a married couple, and are likely a cameo appearance by relatives of lead designer Cory Stockton.
While Jaina is canonically involved with Kalecgos these days, there's a slight hint of attraction between her and Vereesa Windrunner as per the Alliance's side of the Isle of Thunder scenarios. Jaina often dotes on Vereesa.
Wrathion and Anduin in Mists of Pandaria due to how they were frequently seen playing board game with each other in the Tavern while Anduin teasing Wrathion about his age and size as well as the fact that Anduin seems to be the only person that genuinely like Wrathion. Fan fics written by fans about them hooking up after this expansion and/or confront Wrathion's decision to freeing Garrosh pops up frequently. This doesn't help by their interaction in War Crime, with Wrathion acting like he wants to confess with Anduin right here and there if not for different goals.
Tirion Fordring and Darion Mograine. Tirion saves Darion's life after Artha's steals his father's soul, and instantly partners up with him, trusting him implicitly. In the Ashbringer comic, Darion seeks out Tirion to find out how to free his father's soul from the Corrupted Ashbringer. Tirion's obligatory answer? Love. He doesn't even bother hiding his horror when Darion is turned into the Noble Demon that he ends up as.
Eventually, Darion returns the favor and saves Tirion's life, yelling at him, berating him, kicking stuff around, basically raging that Tirion did not come to him immediately after discovering the heart of the Lich King. Those guys are really invested in the other one living because of pure trust, or...something else. They always talk to each other while ignoring other people. Stand by each other when watching inappropriately timed challenges, etc.
Kama the Beast Tamer, the female Pandaren stable master just outside the Tavern in the Mist, apparently has a female 'life-mate' according to the in-game erotic literature 'A Steamy Romance Novel: Hot and Misty'
Another of the Steamy Romance Novels, entitled 'Savage Passions' features two males engaging in incredibly suggestive acts.
In Warlords of Draenor, the Alternate Universe Drek'thar, like all characters, has several on-click audio clips. One of his 'pissed' lines (the ones you get when you click on them many times in a row) is as follows: "I...didn't know you felt that way about me. Perhaps we should take this somewhere more...private." He will deliver this line regardless of the Player Character's gender.
More on the Foe Yay side, but Kil'jaeden and Velen were very close. In the World of Warcraft novel Rise of the Horde, Kil'jaeden's reaction to Velen fleeing Argus to avoid corruption smacks of jilted lover, and in Warlords of Draenor, Archimonde even calls out Kil'jaeden on his obsession about about getting revenge on Velen (and dranei), and how it aparently slowed down the conqering of Draenor.