I'm having the same problem, tried lots of things to try to solve it and it didn't work. And, after a quick search, turns out this is a widespread issue (examples
http://android.stackexchange.com/que...nt-login-error and
http://forums.androidcentral.com/ask...o-account.html, but there are more).
Some say the issue is because Google servers, but my bet is that this is related to its Google Play Services app, which is the one that auto-updates on my phone. I also remember reading those issues on the recent comments on its Google Play page.
For now, the only way to get ride of that warning icon is to disable Auto-Sync (on Config-Accounts), but your mail and contacts won't sync, of course.
So, hopefully Google fixes this soon, as this is surely annoying :S