Consider a board where the birthday feature is enabled but the guests cannot view profiles (and cannot view the viewonline and memberlist pages, since it's the same permission)...
If guests are not allowed to view profiles, the birthday block always displays the message "No birthdays today", regardless the actual members' data. Therefore, the birthday block should be entirely hidden if they cannot view profiles.
Since phpBB 3.2.0-RC1, the online registered users list is now hidden if you cannot view profiles/viewonline/memberlist. The same template condition should be added for birthdays.