What's known about basing one-way function on the $P \neq NP$ assumption? Is there a conditional impossibility result or the question is completely open?
We cannot hope to prove a general impossibility result since if one-way functions exist (and we believe they do), then in particular it follows that the statement "If $P \ne NP$ then one-way functions exist" is true. 

However, we can prove that certain proof techniques are too weak to prove that statement. In particular, the following paper of Akavia, Goldreich, Goldwasser and Moshkovitz proves that this statement cannot be proved by certain black-box reductions (conditioned on plausible assumptions):

http://www.wisdom.weizmann.ac.il/~oded/p_aggm.html