Doze is the feature in the latest Android versions which puts unused apps to sleep. This can cause an app not to send notifications. You can exclude apps from it.
(I don't have a Z3, but do get alerts from FB. Usually for things I've already read, sometimes on a different device but sometimes on the same one! I think they just don't care, even if they annoy you with crap notifications it's still "increasing engagement" in their minds. I disabled notification badges for the app and am considering disabling notifications completely or ditching it: nothing on FB is important enough to need a notification anyway.
Ironically I have had problems with Messenger notifications, which is the thing you would want to notify you...).
