Do Nirnroots regrow in Skyrim?

Nirnroot does indeed grow back and can also be bought from Alchemy stores. It is a lot less rare than in Oblivion. With regards to the Crimson Nirnroots, I am not sure, because I haven't come across them yet. However, I believe they would re-spawn too. (All alchemy Ingredients I have seen so far do.)

EDIT:

I just wanted to confirm that crimson nirnroots do indeed regrow after 15 in game days, as seen here.

A quick warning: do not pick to many crimson Nirnroots before starting the quest, as it has been reported that this will cause negative progress through the quest.

I can confirm they grow back after 15 days also. I'm using the PS3 version and picked about 20 of them before doing the related quest. It does give you a negative number if you try dropping them and picking them back up. So, I was forced to wait 15 days to pick a total of 50 of them.