Before you update to Google Authenticator 2.0.0 for iOS, disable two–step (or wait until 2.0.1). It removed my tokens from the app, and I had to re–add them again. Boo.
To fix this with a Google account, you’ll need to do the following:
1. Go to Account → Security → Edit (under 2–step verification).
2. Under How to receive codes and across from Mobile application, click “Move to a different phone”.
As for the backup codes you printed, if you click the help icon (question mark) next to “Remove Switch to phone”, it says:
When you want to switch to a different phone, select “Move to a different phone” and follow the instructions to configure the Authenticator app on your new phone. This will not invalidate any of your existing application-specific passwords or backup codes.
Dropbox: In Account → Security, click the Edit link next to Authenticator app.