On the GW6900 atomic solar (3179), button beeping can be turned on and off by going into normal timekeeping mode, and pushing the set button (upper left). Then you press the lower left button to scroll through the various settings, which include backlight duration, power saving mode function on/off, etc. One of them is the button beeping, which comes just after the day setting (last time/calendar setting) and just before the 1 or 3 second illumination duration setting. Looked at the manual for the DW6900 (1289) and verified there is no setting to turn the button beep off. As described by others, the spring contacting the back facilitates the beeping. The "white" thing in the back is the annunciator that does the beeping. Cover over the bare spot the spring contacts with removable tape, or a piece of paper like a small Post-It note, and the beeping will stop, along with all alarms, hourly signal, timer ending, etc. I don't recommend removing the spring as they're all but impossible to replace if lost. Better to insulate the back from it with something that won't rip the annunciator off or damage it when you remove it (i.e. don't use duct tape or Gorilla tape).
John