Yes human nature is obsolete. Ever since we evolved from primates into humans the nature of humans is to advance. The advancement of humans has come to a screeching halt, as we now focus on advancing smart phones and computers to do work for us. We need to focus on advancements that will be a betterment to mankind as a whole, such as advances in space travel and promoting new inventors.
This question can't really be answered until you define "human nature." We, as humans, have the ability to reason and to choose our own actions. However, one could argue that human nature cannot be obsolete, since "human nature" refers to something intrinsic about humans, so once it becomes obsolete, it is no longer human nature.
Human nature is, at its core, about adaptability and evolution. Granted, it's an aggressive and often parasitic form of evolution. The nature of humanity has nothing to do with loving or caring or hating or killing or any other such thing, it has to do with overcoming obstacles. It is no different than the nature of any other animal in our world, we're just the animal who developed denial and excuses.
No, I do not believe and would have to disagree that human nature is now obsolete. First of all, this depends on what you define as "human nature." To me, human nature is caring for other people and other behavior such as that. This kind of behavior or human nature, is absolutely not obsolete.