Snom phones with version 8.x show a dark screen with the text DND when Do not disturb is on.
The Snom D785 and D385 run firmware version 10.x, and in version 10.x this screen is not there by default. You can however implement this with some simple steps.
Click on Action URL settings. Enter the following:
DND on: http://user:firstname.lastname@example.org/settings.htm?settings=save&custom_bg_image_url=http://downloads.snom.net/documentation/snom_screenover_dnd.png
DND off: http://user:email@example.com/settings.htm?settings=save&custom_bg_image_url=