I have a "mini" dongle that I bought for use Cabrio Bob's ADU. I leave it plugged in to the OBD receptacle 24/7. It has been plugged in this way for at least 8 or 9 months with no negative results.

 

Likewise, the OBD connection to my HKS Camp2 stays plugged in all the time.
But prior to getting it, I had the CEL come on while driving, so I plugged in a dongle (still driving) to read the codes. Big mistake! A few seconds later I had a Christmas tree light show and the car went into limp mode. (Pulled over, shut it off, unplugged the dongle and restarted, back to normal, except the CEL was still there, duh.)

 

I leave my Bluetooth in all the time. I've checked and it doesn't add to battery drain. It MAY, however, mess up my TPMS. Not sure, I've not checked that part of it (yet).