In regard to postmodern literature, I think we should see metal as a reaction against a postmodern world. Whereas postmodern art, theory, and literature more or less seek to deconstruct, and to some degree to affirm the banality of the everyday, metal strives to overcome this by hearkening back to ancient times and values. Postmodernism is more a cultural analysis. Meaning, cultural analysis is postmodernism's ultimate aim, whereas metal wants to overcome and ignore culture to find a sense of higher beauty.
That is the key distinction I see: for example, visual art in the postmodern world does away with aesthetics almost entirely; it does not concern itself with 'beauty', and instead seeks to find something to replace it in the everyday, that being some pseudo-philosophical concept that ultimately goes nowhere. In fact, traditional beauty is often regarded as tacky and undesirable in the postmodern world. Looking back to Duchamp, and later Warhol, and now Jeff Koons and Damien Hirst, we see the trend has moved toward appreciating commercialized products, or indistinguishable replicas, as beautiful objects, either for their aesthetic appeal or their shallow cultural criticism. There is no content, really, but people can be convinced to appreciate them and accept them as art.
In any case, what I have briefly described above is pop art, which has deep ties to 'conceptual art, which we might want to regard as a counterpart to liberalism and hipsterdom. But look at the term: pop art. It is the same as pop music, a result of a valueless, postmodern world in which we seek to appeal to the lowest common denominator not by creating something beautiful, but by creating a synthetic beauty to appeal to the masses. This is not the goal of metal, or any greater art.
I will acknowledge that postmodernism has its interest in the darker side of things (nihilism, atheism, death, and so forth), but it is always linked to a sarcasm that is absent in metal. If we can see any bands related to this sort of ideology, it would be those such as Deicide and Slayer and other death metal acts who mock the traditional symbolism valued by Christianity. In black metal, however, as well as a good amount of death metal, the quest for beauty, not to mock the valueless and the mundane and thereby catch itself in it, but tocreate value in a world which is inherently valueless, is what qualifies it as something more than 'postmodern music', though it was created in the same era.
In short: yes, metal is certainly influenced by postmodern literature, but we should see its inability to escape this contemporaneous ideology as its downfall.