Android - "Show notifications" is greyed out for some apps. What's causing it and how do I change it?

With root access, there are some apps such as 3c Toolbox that allow you to change the notification settings for any app. App Ops is another one that gives you control over all apps' general permissions.

You can actually manage all app permissions from the command line with root access. In Android 5.x, appops was added. It gives control over general app permissions.

Find your app's package name by looking through the list generated by:

# pm list packages | sed 's/^package://'

Get a list of permissions and their mode (allow/ignore/deny/default) for a particular app:

(Note that get parameter is not available in 5.0.2 or earlier Lollipop releases, but was introduced in Android 5.1.0)

# appops get <package_name>

Set a particular app's permission:

# appops set <package_name> <permission> <mode>

To accomplish the same thing as unticking the "show notifications" box in the application manager:

# appops set <package_name> POST_NOTIFICATION ignore

From my experience, using deny as a mode can break appops so that permissions can't be accessed or changed from an app, so I recommend using ignore, which is what the system uses.

If in some case, you end up breaking access to app ops through something like the application manager, you can reset the permissions for an app:

appops reset <package_name>

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For more control over detailed permissions, you can use pm in the terminal. This works with older Android versions.

Find your app's package name by looking through the list generated by:

# pm list packages | sed 's/^package://'

Then list the permissions for a particular package:

# pm list permissions <package_name>

Find the notifcation permissions:

# pm list permissions <package_name> | grep NOTIF

Grant or revoke a particular permission:

# pm grant <package_name> <permission>
# pm revoke <package_name> <permission>

The permissions listed with package manager are very detailed. Using a capable app to manage them may be a more practical route.

jan's answer already addresses how to change the notification policy for such (disabled "show notifications") apps.

As for why and when (or essentially, in what condition it applies), your guess of system apps is on the right track.

To be exact, as Sergey has explained in the comment, only packages that are signed with the platform key (from vendor/manufacturer, e.g. AOSP) will have this privilege. These packages are part of the core platform. Other preinstalled apps (including apps with some special privileges, which are signed with the shared or media key) and "system" apps (apps which are put on /system/app folder and granted access right) are not counted.

Summary: as of the latest version (Android 5.1.1 Lollipop), the checkbox is disabled if:

1. The app is signed with platform key (part of the core platform), or
2. The app is not installed on the current user (in multi-user case)

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From InstalledAppDetails.java (this is what "App Info" displays),

private CompoundButton mNotificationSwitch; // this is the "show notifications" checkbox

...

private void initNotificationButton() {

    ...

    if (Utils.isSystemPackage(mPm, mPackageInfo)) {
        mNotificationSwitch.setEnabled(false);
    } else if ((mPackageInfo.applicationInfo.flags & ApplicationInfo.FLAG_INSTALLED) == 0) {
        // App is not installed on the current user
        mNotificationSwitch.setEnabled(false);
    } else {
        mNotificationSwitch.setEnabled(true);
        mNotificationSwitch.setOnCheckedChangeListener(this);
    }
}

Note:

• Utils.isSystemPackage() returns true only for packages signed with the platform key.
• This doesn't apply on Android 6.0 Marshmallow due to overhaul on App Info's user interface; there's no checkbox, instead there's a centralized notification settings per app. However, the underlying principle is still the same that you cannot block notifications on an app that is system package.