The damped string stops sounding. Nevertheless it is important to remember that it's resonance can live on in the harp.
Single string damping.
If the string should be damped immediately after it is played, the note can be notated as a staccato. If the damping is delayed, however, a damping sign is placed after the note.
Delayed damping of the string is notated by placing the damping sign visually in the score or by writing small notes that have a damping sign on their stem. For rhythmic accuracy the second option is preferable.
Single string damping. Visual/rhythmical.