I was just wondering, is there a project like this?
Maybe there is but I didn't hear about it.
What do you think about that?
Stack Exchange Inc (SEI), in and of itself, can be considered a social responsibility project. The site not only provides a place for people to get answers, it does so without charge, and even goes so far as to give away, via creative commons, the data that is collected from its users.
The mechanisms SEI uses to encourage users to contribute their time and effort for free end up encouraging users to provide a wealth of information to the world.
In a sense, Stack Exchange provides a place for people to fulfill their individual social responsibility, and does so in a manner that people want to contribute to this global knowledge store.
So not only does Stack Exchange fulfill its social responsibility, it gets individuals to fulfill theirs.
As pointed out by @balpha in the comments, they have made donations to several charity and Open Source projects in the past.
they also sponsor the Bad Code offsets project if you count that as a "social responsibility project".