Disable Plugin Update Notifications
remove_action('load-update-core.php', 'wp_update_plugins');
add_filter('pre_site_transient_update_plugins', '__return_null');

Disable Theme Update Notifications
remove_action('load-update-core.php', 'wp_update_themes');
add_filter('pre_site_transient_update_themes', create_function('$a', "return null;"));

Disable Core Update Notifications
if (!current_user_can('update_core')) {
            return;
        }
        add_action('init', create_function('$a', "remove_action( 'init', 'wp_version_check' );"), 2);
        add_filter('pre_option_update_core', '__return_null');
        add_filter('pre_site_transient_update_core', '__return_null');
    }