I have ~8 logins that use Google Authenticator (Google, FB, Bitcoin stuff, etc). I made a backup of my Authenticator app/sqlite database before moving to my new phone. How do I restore the database/settings on my new phone without root?
There's a walkthrough on exactly this topic here: http://www.howtogeek.com/130755/how-to-move-your-google-authenticator-credentials-to-a-new-android-phone-or-tablet/
Essentially, it amounts to disabling two factor authentication and re-enabling it, or for services like Google, using their "replace old 2 factor with new 2 factor" service. It is also possible to extract an sqlite database of your credentials, if you have adb and can access the phone that way, which I believe is possible without root.