Storytelling in massively multiplayer games usually occurs only in footnotes. You might read a lore item's description here, get a hint in the quest text there, but it's almost always an ancillary part of the experience. In the olden days of text MUDs, that wasn't necessarily the case. In some MUDs, players and wizards engaged in communal storytelling, as in the best pen-and-paper roleplaying sessions.
If you look at today's mainstream online games, it seems as if that art has been lost. There are some smaller communities out there that still herald that kind of experience, though. The Escapist focused on one of those in an article titled "The Ending Has Not Yet Been Written." It's an excellent piece about how players and developers alike have fostered a unique, niche-storytelling experience in Cyan Worlds' Myst Online: Uru Live.
Myst has always been an eccentric in the gaming world. It was a groundbreaking success for computer games, and it spawned countless clones, but no one ever recaptured its magic. Now the series is treading a unique path in the online world, despite past setbacks.