Dating Catwoman

The Trope Namers Batman & Catwoman are at it again in Batman: Arkham City, providing the trope image. There wasn't quite as much flirting in their first meeting in Batman: Arkham Origins Blackgate, but you can tell where it got its start. By Batman: Arkham Knight, Batman decides it wouldn't work out, due to his decision to give up being Batman. He does sound genuinely heartbroken over it.

Something similar happens in Injustice 2. In Catwoman's arcade mode ending, she enters a committed relationship with Batman, but being plain old Bruce and Selina doesn't excite her as much; their relationship only thrilled her when it was Forbidden Fruit. Also, being a billionaire's girlfriend meant she'd never have to steal again. Getting everything she wanted bored her senseless. She ultimately leaves him to return to her old life.

In Batman: The Telltale Series you can romance Catwoman as Batman and even have sex with her at one point in Season 1. The Golden Ending of Season 2 has Batman tell her he loves her and secure her freedom from Amanda Waller and the Suicide Squad, but with the bankruptcy of Telltale Games it's unlikely we'll see the relationship progress further anytime soon.

Sly Cooper and Carmelita Fox. He's a heroic Gentleman Thief, she's a dedicated police officer. It's kind of one-sided to start with, but she softens up in the climax of each game, then gets really angry when Sly slips away. At least until the third game, where they actually hook up for good at the end. Sly willingly gives up his life of crime and all of his friends, saves her life, and fakes a convincing amnesia act to let her justify it. Then in the next game, it's back to business. Sort of.

Turns out that your love interest in Quest for Glory IV is a powerful vampire known as The Dark Master who wants to resurrect the local Eldritch Abomination.

The situation with Silk Fox and the player (either gender) in Jade Empire starts out looking like this, before a variety of revelations related to the Gambit Pileup point out that she is most definitely a good guy.

Knuckles and Rouge from Sonic the Hedgehog fit this trope very nicely, especially in Sonic X.

Alex, the only real Love Interest in The Bastard of Kosigan series is also the brains behind one of the main villain's operation. Though, due to the story's Black and Gray Morality, she shows up much more on the grey side of the scale than the French, the Inquisition, or the witches, so perhaps she and Mordred aren't actually villains after all.

In Harvest Moon: Magical Melody you can marry your rival, Jamie, who is always the opposite gender from you. You and your rival are implied to be the canon pairing. Sadly, however, your game ends if you marry them, and you can't even find all the magical notes. Harvest Moon DS Cute also provided the option of marrying Gentleman Thief Skye.

MegaMan Volnutt and Tron Bonne in Mega Man Legends.

A Steamy Romance Novel: Forbidden Love is a Vendor Trash item that can be read in World of Warcraft that details a romance between Marcus, an Alliance paladin, and a blood elf, the book clearly indicating that she's a loyal member of the Horde.

Leon Kennedy and Ada Wong of the Resident Evil series alternate between teaming up and pointing weapons at each other. All while oozing so much Unresolved Sexual Tension into the air that it could be cut with the proverbial knife.

Assassin's Creed:
In Assassin's Creed, the Assassin Altair and the Templar Maria end up falling in love, despite being enemies. This is shown in Assassin's Creed II, where Desmond even has a memory of their child being conceived, and Altair's Codex contains a hand-drawn portrait of her. Worth noting that Maria actually switched allegiances in Assassin's Creed: Bloodlines, and canonically the romance came afterwards.

In Assassin's Creed: Unity, Arno Dorian's main love interest is his childhood friend Elise, a Templar who is the daughter of the recently-assassinated Grandmaster. He ultimately ends up choosing her over the Assassins, which leads to his mentor Bellec attempting to kill them both and him being expelled from the Brotherhood.

Otacon and Sniper Wolf in Metal Gear Solid. One-sided from him to her.

Dragon Age: Origins:
The Warden and Morrigan border on this: while on the same side, she's one of your more ruthless teammates and has her own agenda for following you. Wynne mentions in party banter that she had a son. The Asunder novel reveals that the father was a Templar. World of Thedas: Vol II heavily implies that the Templar was Knight-Commander Greagoir.
Mage!Hawke and Fenris, or Templar-aligned Hawke and Anders or Merrill in Dragon Age II.
The Ship Tease between Templar!Carver and Merrill, as well as Circle!Bethany and Sebastian Vael. Rival Romance in general can be considered this trope, since Rival Hawke is everything opposite of their love interest. If neither are romanced, Dorian, a Tevinter Mage, and The Iron Bull, a Qunari from Dragon Age: Inquisition will be this.

Garen and Katarina in League of Legends, who quite literally got matched via a dating service, yet are champions of opposing countries. They are Not So Different not just personality-wise but in gameplay, being two of the three champions that operate solely on Cooldowns with no Mana Meter, and having signature moves that involve lots of spinning.

Mass Effect: Commander Shepard (Alliance) and Miranda Lawson (Cerberus) in Mass Effect 2.
All but outright stated between Aria T'Loak and Nyreen Kandros in the third game's Omega DLC. Aria is the cold-blooded ex-merc whose word is law on her Wretched Hive, Nyreen's ex-turian military and always thinks about the little people ("she oozes virtue"). It ended when they realized it couldn't work unless one of them changed, and Nyreen didn't want that to be her.

And yet another BioWare case: In Star Wars: The Old Republic, Lana Beniko (a Sith sorceress) and Theron Shan (a Republic spy) can be romanced, regardless of faction or gender of the Player Character. A Jedi dating Lana or an Imperial Agent flirting with Theron would fall into this category.

In The Elder Scrolls V: Skyrim, the Dawnguard DLC has a subtle Ship Tease between Serana of the Volkihar Vampire Clan and the Dragonborn, even if the latter chooses to remain aligned with the Dawnguard. However, Serana cannot be married unlike other follower characters, meaning the relationship never really progresses beyond platonic friendship.
Farta the Troll loves her masochistic counterpart, Eryi the Fairy, in Eryi's Action.

In Infernal, the banter between Lennox (the protagonist) and one of EtherLight's people, Major Elena Zubrov, implies this kind of relationship, but in reality, that's just because Lennox is being impertinent and Elena is playing along. It certainly doesn't turn into anything (given that Lennox eventually kills her and eats her soul).