Q:

Can we run firebase realtime listening on ios serviceworker? 

As of iOS 11.3 there is serviceworker feature supported
Even though it has no support for notification.  I wonder if it possible to just run any realtime listener, such as firebase database, to receive realtime data and use ServiceWorkerRegistration.showNotification to show local notification instead
Is it possible? 

A:

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.