Something the comics have never answered expressly, an you could make arguments for either side, but I don't believe Wolverine's adamantium grows back.

First, yes it's beta adamantium which is stated to mean that it does not inhibit his bone's natural functions, but never once is it mentioned to actually heal. You would think the writers would toss that in if it's something they intended.

Second, Wolverine's adamantium has been damaged, and not healed. In Inner Fury, nanobots fucked with his adamantium, and his healing did not restore it (in fact his healing made it worse by trying to eject it). Similarly, Old Man Logan had his hand cut off and it regrew as bone. We also see lots of future timeline wolverines (even future 616 wolverines) who have lost limbs and not regrown adamantium.