Superheros are exciting and cool because they are special, relatively rare, and interesting in a meaningful way. An MMO where everyone can be a superhero completely destroys each of those points: superheros become plentiful, mundane, and end up performing repetitive tasks.
Andrew has an entirely valid point. This is, however, a sub-genre of superhero stories. You do not see them often, and far less often well thought through, but you do see Astro City and others that take the notion of having a city of superheroes. I wish I could remember more, but my reading of comic book deconstructions is way behind. There have been comics about the equivalent of superhero internal affairs and the clean-up crews that deal with all these heroes. Other comics occasionally toy with the idea, like Silver Age stories where there are entire cities of Supermen or JLA Rock of Ages.
I will cite this last for how it is hard to do well, because while Rock of Ages was a great story arc, the DC universe then politely ignored the social implications of literal angels appearing on Earth or temporarily granting the entire planet superpowers. Or even what happened in those hours of ubiquitous demipowers.