When Spock was being rapidly regenerated on the Genesis Planet in Star Trek III: The Search For Spock, why didn't his hair and beard grow rapidly as well since he was rapidly being regenerated? Is this a movie plot hole or am I missing something?
One plausible explanation - and this is largely speculation - is that the Genesis process wasn't simply rebuilding Spock's body from purely DNA, but was somehow using his corpse as the template. Given that we know so little about how the Genesis device works, and especially how it interacted with Spock, it's entirely possible that it was restoring Spock to a previous condition - kind of like a backup copy.
What little evidence I have to support this is found within "Star Trek III: The Search for Spock" itself. Since Spock was found as a child, many fans have assumed that the Genesis process was rebuilding Spock from his Vulcan DNA. Although this would explain why he went through the aging & growth process again, it does NOT explain what happened to his original body. If Genesis were simply building him a new body (which would have hair growth as you stated), then his original body would still be inside the torpedo casing found by Dr. Marcus & Saavik. Instead, the corpse was gone and the microbes on the outside of the casing had rapidly evolved into new life-forms.
Based on the fact that it appears to be Spock's original body "reset" to a period of childhood, I believe that the Genesis effect was actually restoring him to his previous state - as he was right before death. If this is true, and granted that's a big IF, it would explain why his hair returned to a length and state similar to before. It would also explain why he rapidly aged to his previous appearance, then began aging normally again.