Quests already are a small part of online multiplayer progression (CoD Pro Perks, Battlefield Assignments etc.). They could potentially take over a much bigger role. I believe, if done right, quests could be an amazing alternative to traditional leveling. It allows for true prestige equipment, and doubles as a means to introduce players to more and more advanced playstyles, as well as promoting more diversity in weapon and gadget loadouts.
I think we'll see a push in this direction within the progression system of the next iteration of Battlefield. If it sticks, I think most shooters will follow suit.