Via Dan: Will redheads be extinct by 2100? (Or, at least, extinct in terms of new redheaded children being born.) Barring intervention, it really is just a question of when, and offhand 2100 doesn't seem implausible to me. A question is, as red hair becomes rarer, will those remaining be want their kids to have red hair, or want their kids not to have red hair? If the latter, dissortative mating could bring the end of the auburn world as we know it even sooner.
My guess is there will be more of an effort to save blonde hair and, even more, blue eyes.
I have nieces with red hair and blonde hair, but the six identically brown-haired kids produced by my parents would lead me to suspect only one parent has recessive genes for hair lurking within (my father, since I have redheaded cousins on that side). My own brown eyes seem weirdly like they have become a bit greenish in recent years, which I've attributed to the DNA effects of the unfortunate latch on my microwave.