The moral case for non-anthropocentrism is far stronger than for the contrary -- even from the point of view of a self-interested human!
Anthropocentrism isn't just bad for the trees, birds and algae that, through no fault of their own, have to share the planet with us -- it's bad for us humans too. It's a narcissistic ignorance that stops us from really taking care of ourselves. Or, if you like, genuine humanism needs to be non-anthropocentric.
It's like growing up: to become an adult you have to take responsibility not only for yourself but for others too -- and you do this not just to be nice, you do it because that's the only way anyone will do anything for you. Becoming an adult means engaging in a web of interdependence and breaking from the wonderful but naive narcissism of childhood.
So it is with the anthropocentric myopia. It seems as though it's the most human, most moral of mindsets but actually it's naive, a distraction. Children lucky enough to have caring parents don't have to worry about how the care of themselves presupposes their care of others. Humans aren't so lucky -- God's dead, after all.