I'm also experiencing the same thing with my DAC III. I leave it on all the time but when I turn on my cd player which I use as a transport, it clicks. Same thing when I turn the cd off or when I unplug the dac. Similarly, I hear it with my cd player (rotel rcd 970) when it is turned on.
I just thought that this was normal because the relays are switching something on. Birdieman's explanation made things clearer for me too.
I'm wondering whether moving the jumpers could also affect/improve the sound.