If you are already running Windows 10 Mobile Build 10166 or higher you can go to Battery Saver settings by following System–>Battery Saver–>Battery Saver settings. This setting was present on Build 10149 too but was not working and used to crash on trying to open. But on Build 10166 it works and allows you to add or remove individual apps that can run in background and also push notifications, even when Battery Saver is on.
As you can see in the above screenshots when you tap on Battery Saver settings it takes you to “add an app” screen. Tapping on add an app throws a list of apps that can be added here. If you want to remove an added app you can again tap on added apps and then remove. This really provides with some flexibility for users as one can really choose to allow apps to run in background and also push notifications if required.