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Nice question. I really like it but the truth is, it's almost impossible for mankind to go extinct. While in the beginning, man's evolutionary path led to a few stages of morphological changes, the major differences between the homo erectus, habilis, sapien, etc is the increase in overall intelligence. Mankind has developed physically over all the years that we've existed but unlike other animals, we've gotten smarter as we've evolved. And that increase in intelligence is what has put us on top of all other animals.
Now for what I said earlier. It's almost impossible for man to go extinct and this is because of our intelligence. Mankind is capable of problem identification and we are very much capable of problem solving. If ever a global problem arose that could make us go extinct, we'd find a solution to the problem. And if by chance it was something that would ravage us totally without us being able to do something about it, then it would probably have the same effect on animals too. Basically I'm saying that anything that will bring about the end of mankind will equally bring about the end of other animals too.
So in essence, unless Thanos personally snaps us away, then we're not going anywhere for a very very very long time