It's my understanding that the chirping is the DAT tape "spinning" in the drive. In other words, it is the background noise the correlates with the RPM of the tape passing over the "head" or "pickup", to use an analogy from the old days of analog casette tapes.

It is a constant background signal, meaning that as you turn up the volume, this chirping does not get louder and the music actually drowns it out. If you turn the volume all the way down, the chirping becomes more apparent, because the music is not there to hide it anymore.