Disable Please visit the Update Network page to update all your sites. Notice from Dashboard

This message was fired on two hooks.

add_action( 'admin_notices', 'site_admin_notice' );
add_action( 'network_admin_notices', 'site_admin_notice' );

Remove this and you remove the message, use remove_action.

To remove this function use a function, like the follow example.

add_action( 'admin_menu','fb_hide_site_notice' );
function fb_hide_site_notice() {
    remove_action( 'admin_notices', 'site_admin_notice' );
}

Now for the network ares:

add_action( 'network_admin_menu', 'fb_hide_network_notice' );
function fb_hide_network_notice() {
    remove_action( 'network_admin_notices', 'site_admin_notice' );
}