How do you disable the admin panel Inbox Notification pop up?

Method 1

Under:

System > Configuration > Advanced > Advanced > Disable Modules Output simply set Mage_AdminNotification to “Disable”.

Method 2

The previous method works fine but potentially, a user can re-enable these easily. To make the change a bit more difficult to undo, we can disable the module from the modules XML files. If we already have a custom module, we can simply add:

<Mage_AdminNotification>
    <active>false</active>
</Mage_AdminNotification>

To our module XML file. A complete example could look like this:

<?xml version="1.0" encoding="UTF-8"?>
<config>
    <modules>
        <MageBase_Custom>
            <active>true</active>
            <codePool>local</codePool>
            <depends>
                <Mage_Core />
                <Mage_Cms />
            </depends>
        </MageBase_Custom>
        <Mage_AdminNotification>
                <active>false</active>
        </Mage_AdminNotification>
    </modules>
</config>

Reference LINK

From http://www.nyssasutherland.com/how-to-disable-annoying-magento-admin-popup-messages/

Simply login, go to System -> Configuration -> Advanced and change the “Mage_AdminNotification” drop down to disabled.

If you do not want to disable the message feature completely, You should not disable Mage_AdminNotification.

To just disable the annoying popup you should rewrite Mage_Adminhtml_Block_Notification_Window. File path is app/code/core/Mage/Adminhtml/Block/Notification/Window.php. You should override canShow() function like this:

class Your_Module_Block_Notification_Window extends Mage_Adminhtml_Block_Notification_Window
{
    public function canShow()
    {
        return false;
    }
}

Here you can find more on overriding in magento.