Yup, Autotune *should* take care of everything when you switch domains. Obviously, re-test everything on live, but that's the purpose.
The other options are:
* Use Autotune on your dev site, which will setup all the sub-domain URLs properly. Then, go to the Facebook Developer Area, edit your app and add all the 'live' URLs to wherever you see the dev URLs. That should make the app work on both sites.
* Create a new app ID for the Live site and re-run Autotune. That will keep your dev and live sites separate from Facebook so you can test new user registration and other features with the same user.
I hope that helps,
Alex