David Foster Wallace is often called a postmodern writer—sometimes even a post-postmodern one, but his relationship with postmodernism is conflicted. 

What is postmodern about him is that he keeps a cynical distance from his material. He refuses to take anything--even his own written descriptions--as innocent or without an ideological slant. Like postmodernists, he makes frequent use of irony, juxtaposition and sarcasm with the intention of leaving readers to make up their own minds.

Wallace, however, yearns for sincere emotions in a way that postmodernists did not. Whereas postmodern writing was ironic with the purpose of breaking down established truths and showing their falsity and meaninglessness, Wallace wants to do more. He seeks to make a new kind of meaning arise through the ways he leverages the different, always incomplete and patially false, perspectives he presents in his works. 

Here’s how Wallace answers the question, himself:

The problem is that…what’s been passed down from the postmodern heyday is sarcasm, cynicism, a manic ennui, suspicion of all authority, suspicion of all constraints on conduct, and a terrible penchant for ironic diagnosis of unpleasantness instead of an ambition not just to diagnose and ridicule but to redeem. You’ve got to understand that this stuff has permeated the culture. It’s become our language; we’re so in it we don’t even see that it’s one perspective, one among many possible ways of seeing. Postmodern irony’s become our environment.

Source: Larry McCaffery’s “Conversation with David Foster Wallace” (Dalkey Archive Press at the University of Illinois: Summer 1993).