A Church Account is not required to use Gospel Library. However, there's an authentication bug in iOS Member Tools 4.8.0 (released last month) that affects other Church apps (Church apps use single-sign-on), where signing in to Member Tools can cause other apps like Gospel Library to get into a partially-signed-in state (signed in but not fully signed in). Until Member Tools is updated (hopefully in the next week or so), the fix/workaround is to go to Settings in Gospel Library and fully sign out (which clears the bad state), then sign in again.
There's no need to sign in if you're just reading content, but signing in is encouraged if you use any of the note or annotation tools, like notebooks, study plans, highlighting, etc. – signing in backs up the annotations so they're not lost if something happens to your app or phone and the app needs to be installed again. Anyone can create a Church Account – member or nonmember – at Account.ChurchofJesusChrist.org.
Samuel Bradshaw • If you desire to serve God, you are called to the work.