If you are attempting to log in to Facebook on a Beta version of the Mevo app on iOS, but are stuck in an authorization loop, it's likely because Facebook is trying to authorize the production version of the app (i.e. the version you downloaded from the App Store) rather than the Beta version.
To resolve this, delete the production version of the app from your iOS device.
After it is removed, open the Beta app and log in to Facebook again. This time, once you allow Mevo's permissions within Facebook, Facebook will redirect to the Mevo Beta app and you will be logged in.
Once you have successfully logged in to Facebook on the Beta app, feel free to re-download the production version of Mevo from the App Store.