firebase is a storage. You wouldn't expect storage to do any actions.
For local notification generated by app when it is in the background you can constantly listen to changes in firebase and when the change occurs you can generate your push notification. In order to save the connection time since you are charged by concurrent connections, you can go offline and go online every 5 min for 20s. During this time firebase sdk connects to server, collects all the changes and runs all callbacks. You decide within your app what to do.
If you would like to enjoy remote push notification. You would need to setup a server listening to changes on firebase and talking to APNS which will push the notification to client's device.