For months I did not have the Tesla app open on my phone and everything worked fine. Suddenly in April the car does not know me unless the app is open on my phone.The Tesla app must indeed be running on your phone. It doesn't have to be in the foreground obviously, but it needs to be running. It is the app that handles the "handshake" between the phone and the car, that authenticates you as the owner. The Tesla app should not be doing any background communication or other processing so I sincerely think it will not drain any significant power from your phone battery. Anyway, that's how it works. If you want to use your phone as key, the Tesla app must be running, it's a prerequisite.
Something strange in Bluetooth as I can no longer play music from my phone. I select my iPhone as the source and get endless loop.