Can we run firebase realtime listening on ios serviceworker?

While iOS don't support Web Push notifications yet(as of Apr-2018), all you can do is have an event listener for data from server side(Firebase in your case) and receive that data to show it as an Notification/alert or how ever you want to present inside your application(not in iOS notification area).

Unfortunately, this is the best way you can handle notifications for iOS. I know, its not so great for developers :) Hope Apple speeds up on PWA. They are more than a year late to Service worker party itself. What to expect!

Update : One more link on WhatWebCanDo.today and one from Apple which says iOS-Safari don’t support push notification as a web standard defined by w3c. But looks like there is a OS X safari proprietary solution available.