According to the documentation for the Windows Dynamic IP Updater client, there is a Hidden Mode:
Even if you close the program window, it shows its icon in the system
tray area of Windows task bar. You can make the program even more
hidden by choosing "Run hidden" menu item from contextual menu in
system tray icon.
When running in hidden mode, you can see status window by launching
the program from Start menu.
You may also want to consider Disabling Error Notifications:
When an error happens, the program will notify you by showing status
window with error message describing the problem. Sometimes this might
happen often and be distracting. You can disable those notifications
by clicking "Stop nagging me" link present after error message. You
can re-enable notifications by clicking "Start nagging me" link.
Please note that by disabling notifications about problems you might
miss a configuration problem.