Does the PolyTune Clip has an automatic turn off or do I have to do it myself?
No, you don't need to turn it off. The PolyTune Clip turns off by itself after three minutes regardless if there any signals or not. Thus you can you can leave it on your guitar (or bass) head stock all the time and not worry about draining the battery.
Here you will find the PolyTune Clip user manual