Not true. My oldest brother is a dark brunette (what he has left, that is) but the one time he tried to grow a beard, it was definitely red! I'm the only one of 4 kids that actually was born a redhead but my youngest brother (blonde) passed the gene on to two of his four kids.
Same thing with my brother except is was just a moustache he grew. Looked weird as hell but there you have it. I guess just the bottom of his face is Irish.