Arya and Gendry in Game of Thrones (the show). Gendry was given way bigger a role in the series than in the books.
Farnese and Serpico from Berserk. Granted, they're half siblings, but the other one doesn't even know, and the other is almost emotionally dead.
Barristan Selmy and Daenerys have some pretty sweet dynamics in A Dance of Dragons, and at no point is any of it even hinting at any romantic interest from either side.
Harry and Hermione from Harry Potter.
Mustang and Hawkeye from FMA: Brotherhood had a relationship that occasionally hinted towards a romance, but it never becomes that.
Holy Shit, almost forgot about Brienne and Jaime in ASOIAF. That's probably my favorite entry on this list.