I can think of two things that argue for turning the gear off, to balance against the many reasons to leave it on:
1) Heat accelerates the aging of all components, so there is a balance between the turn-on/turn-off wear and the heat wear. At one time, conventional wisdom was that one on/off cycle was equivalent to 30-60 minutes of operation. I imagine that is still true today, with a huge uncertainty as to the actual time interval - it's probably more than a minute and less than a day!
2) We take safety seriously, and our gear is designed with healthy margins. But because Bottlehead gear is not assembled under controlled conditions, there is no way to be certain of its safety. This is the reason it is not, and cannot be, UL rated. I never leave anything running when I am not there, and I advocate that policy for everyone.