Just to confirm:
1) You're using the same user name and password for both LDS.org and the Gospel Library? (Sorry, have to rule out the obvious first.)
2) Is the LDS.org account configured with his membership number? (Don't know that this would impact things, but just in case.)
3) Does _anything_ sync? (Highlights, tags, links, notes on highlights, regular journal entries, etc. - I'm trying to figure out does the sync function fail, or is it just the Duty to God part that fails.)
4) Have you tried going in reverse - entering something in the GL Duty to God area to see if it shows up on LDS.org (best to do a backup first, if that's possible, cuz if it'll only go one way (GL > LDS.org), it might overwrite what's already on LDS.org with blanks from the GL.
A variation on that would be to highlight a scripture, tag it, note it, link it to a second scripture, and then sync to see if that kind of thing will go from the GL to LDS.org (and if you haven't already tested the other direction, try that too).
If this doesn't get us anywhere, the GL developers are good at responding to items sent to them through the in-app feedback, so you could try that. (They aren't necessarily here in this forum, which is for user-to-user support rather than formal tech support.)
FWIW,
Liz