If I remember my "tuning and temperament" class, the detuned pairs of strings don't actually create overtones, but they do create what's called "beating." When the strings are just slightly out of tune with each other, the sound waves go out of phase which creates a pulsing or "shimmering" effect. Detuned pairs of instruments are also an important aspect of the tuning of Indonesian gamelans, one of the things that gives that music its distinctive shimmer. Piano tuners are trained to count the number of beats they hear, which is how they can adjust the piano's equal temperament (if a piano was tuned to a natural or "just" intonation you would hear no beating, but it would only sound "in tune" in certain keys)