If you remove a database from an AG, it is no longer accessible via the listener.
You can suspend synchronization while the application upgrade runs, and can still connect to the listener. That way you don't pay a penalty in sync traffic, but the log will still grow on the primary. Once the upgrade completes, resume synchronization and let the secondaries catch up.
If you change out of the full recovery model to prevent log growth, you'll have to resynchronize with a new full backup and restore, as Jeff mentioned, since the log chain will be broken. This will all take longer.
Suspending sync also leaves you the "all else fails" option; you would then remove the DB from AG, and RESTORE WITH RECOVERY on the secondary replica node to pick up the database state prior to the upgrade.
I do DB deployments all the time with the AG syncing, it takes longer sometimes but we are always online. I also make database snapshots prior to upgrade as an additional fail-safe.