In the original cartoon, we have the male and female Autobots. Given that they're soldiers and there's a war going on, it comes with the territory.
In Transformers: Beast Wars, Blackarachnia and Silverbolt and Airazor and Tigatron are this. The former flirted while fighting each other and then she came over to his side. The latter spent quality time together during patrols in the wilderness. In Transformers: Beast Machines there's Strika and Obsidian.
Billy and Canyon were shown to be this in the episode "Billy's Bucket List". Finn and Flame Princess almost became like this in the episode "Vault of Bones", but near the end when Flame Princess suggests going on a dungeon crawl again some time, Finn replies "Maybe we should just go to a farmer's market next time".
Green Lantern and Hawkgirl discuss dinner plans while on patrols or stake outs. In fact, Hawkgirl's idea of a date is starting bar fights.
Defied example; to Wonder Woman's dismay, not her and Batman. The Dark Knight has gone on record saying he doesn't like romance between team members. Though regardless of Bruce's words, him and Diana work well together◊
Aang and Katara are easily this; although they don't officially become a couple until the final scene of the series proper, the amount of times they've fought together side-by-side, saved each other, fought alongside the rest of their True Companions, fought the same battle separately or independently, did other war-related stuff together, trained together, supported each other and, generally just went through Aang's entire journey of ending a hundred-year war together before that point, can at least be tallied up to half the number of episodes in the series (61). And that's not even counting the comics.
Sokka and Suki have this as Character Development. When they first met, neither took the other seriously as a warrior; him because she was a girl and her because he lacked training. On their second meeting, Sokka was super overprotective of her because of what happened to Yue. On their third meeting, both of them have matured enough to fight as equals, and have shared several kisses.
Korra and Mako become this by the end of season one, with 'Endgame' showing off this trope beautifully in tagteam against Amon. Although the battle comes just before the couple, the feelings were already on the table. Subverted in that they don't remain a couple, as they decide to stay as Just Friends in season 2.
Similarly Korra and Asami only get together at the Grand Finale, having already fought together. "Turf Wars" focuses on them becoming more of this trope.
Varrick and Zhu Li, briefly. Though they've mostly worked on the tech side of things, they go into the field against Kuvira's giant mech right after he proposes to her.
Varrick: Now let's go attach these barely functioning rust buckets to a giant killer smashing machine!
Zhu Li: It's exactly how I always pictured our engagement!
Miss Martian and Lagoon Boy take water based missions together. The latter enjoys all this time alone with his 'angelfish'. Miss Martian's previous boyfriend, Superboy, was pissed off by her brutal battle tactics and broke off their relationship.
Kid Flash and Artemis were this for a while. Then they became Retired Badasses. Then they treat a save-the-world mission in Paris like a date.
Trevor and Sypha in Castlevania are a Battle Couple multiple times both Season 1 and 2, even before they discover they actually like each other. The highlight is during the Final Battle where Trevor and Sypha have one of the most awesome examples of Back-to-Back Badass ever put to screen.
The Venture Bros. has The Monarch and Dr. Girlfriend, later known as Dr. Mrs. The Monarch. Although most of the time Dr. Girlfriend is shown as far more capable than her leader/boyfriend/husband, when it comes to straight fighting they complement each other well.
Enzo and AndrAIa of ReBoot's second season. On the rare ocassion that they aren't busting viruses or traveling the Net va Game Cubes, they can be seen on a date.
Kevin and Gwen from Ben 10: Alien Force. If someone touches one of them, the other will either run to their side or beat up the bad guy (usually then running to the other's side anyway).
Kim Possible and Ron Stoppable. He starts out as her sidekick, but they upgrade to an actual couple and he gains his Monkey Kung-fu powers. In the finale, he ends up saving the world when they face an alien battle couple. In the words of Shego, "The sidekick stepped up!", indeed.
Dave the Barbarian: The parents choosing to be a Battle Couple sets up the premise of the series. They went off on a world tour to smite evil and left their daughter, Candance in charge of the kingdom.
Downplayed with Kid Hero Chiro and Robot Girl Jinmay. They never get to fight together, but they are a couple since the very first episode and ever since Jinmay Took a Level in Badass at the beginning of Season 3, they surely are protective of each other.
Ace Pilot Sprx and Pint-Sized Powerhouse Nova. They usually fight side-by-side, have a few battle combos and if you hurt one of them, the other will kick your ass.
Jack and Maddie Fenton on Danny Phantom are ghost-hunters who fight side-by-side when they aren't at home raising their two children together.
On Total Drama, in the first episode after Jasmine and Shawn hook up, they have to work together to infiltrate the island's secret control center. Their first kiss comes while dodging security lasers, and she's practically swooning when she watches him take out an entire army of "zombie" robotic reality hosts.
Steven and Connie promise to be there to protect each other when danger comes. This is more evident in "Sworn to the Sword" where they both use their shield and sword skills effectively against several Holo-Pearls and even Pearl herself. This continues in "Nightmare Hospital", where they dispatch two gem mutants in a matter of seconds. In "Crack the Whip" they follow Ruby and Sapphire's example by fusing into Stevonnie to fight Jasper.
Aladdin: The Series turns Aladdin's fiancee Princess Jasmine into a full-blown Action Girl, making them a perfect example of a Battle Couple within a team of heroes.
Zeke and Josephine Clench from Samurai Jack are a bit of a subversion in that they used to be this, at least until Zeke filed a restraining order against her. They do team up once more against Jack when they both realize alone they can't hope to beat him, and actually make such a formidable team that they would have won had Josephine not gone back to her old tricks and betrayed Zeke, giving Jack the chance to escape and throw them from the train.
Season five has Jack end up in a Battle Couple relationship himself, with Ashi, the last surviving Daughter of Aku. Their Relationship Upgrade even occurs right after an intense monster fight in episode XCIX.
The Adventures of Puss in Boots: Puss in Boots and Dulcinea gradually become this throughout the series. Although they have their ups and downs, after enough Character Development from both sides, they are fully committed to fighting together as a team by Season 6.