It depends, if you have (the sandboxed version of) Google Play services installed on it then it'll handle push notifications just like on any other device. If it's not installed some apps (e.g. Signal) can run in the background and check for updates on their own, some apps can work with your own push notifications server (such as setting up a UnifiedPush setup), and some apps just won't get notifications if they don't support the latter two.