If the Runner manages to draw through their entire deck does anything special happen? Or can the Runner just no longer use a Click to draw a Card?

I know for the Corp' they loose if they can't draw but what about the Runner. It was suggested by a friend that the Heap is shuffled back in to make a new Stack but after looking through the rulebook I can't find anything on the subject.

Thanks in advance.

Since it is not in the rulebook that you reshuffle then I am not sure why your friend would say that. As per the rulebook, nothing happens at all.

Since the runner is not required to draw they wouldn't automatically lose. However, they are now more likely to flatline form net, meat or brain damage. Neither the FAQ or rulebook mention anything about replenishing your stack in anyway.