I randomly hit this today while trying to run Git garbage collect:
$ git gc fatal: bad object refs/remotes/origin/HEAD error: failed to run repack
I don't understand the ramifications of this, but as suggested in this thread, when I encountered this I just did
$ mv .git/refs/remotes/origin/HEAD /tmp
(keeping it around just in case) and then
$ git gc
worked without complaining; I haven't run into any problems.