This is a great question. It is important to start with a definition of postmodernism.
Perhaps the best way to look at postmodernism is the lack of a grand narrative that encompasses all knowledge. Instead, what we see is the burgeoning of many smaller narratives or discourses. Foucault, one of the great voices of postmodernism, calls this the archaeology of knowledge. There is a relativism of knowledge and objective reality. Postmodernism shows that all perspectives are biased.
With this definition, Facebook fits in perfectly. There are billions of voices. And all of these voices have something to say. This suggests that there is no one dominate narrative or ideology in which people believe. Furthermore, as Facebook and social media grow there will be less of a emphasis on a dominant way of looking at things. With a few click of a mouse and some electronic navigation, a user is able to see many views and perspectives. And he or she will have to decide what to believe, if any.