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