Overtone singing itself isn't very common in barbershop music, but their harmonies certainly make use of overtones. Here's a performance of a song called Play That Barbershop Chord (a stable in the genre). It's not instructive, but it's got some great harmonies: https://www.youtube.com/watch?v=F5bYnRkwpco

Here's a fairly technical article about three different seventh chords, one (the harmonic seventh) being the so-called barbershop seventh chord. There are embedded YouTube videos with computer-generated audio samples. http://www.garygarrett.me/?p=1575