I have had the same problem with the Ring Pro and from the forums it seems all models exhibit this problem. Basically Ring is turning an open loop system into a closed loop system resulting in some current flowing through the mechanical chime at all times.
This current is enough (depending on temperature etc…) to partially engage the chime at a low frequency ( i am guessing 60hz ) resulting in a buzzing sound.
The voltage across chime is 4 vac in this state is about 4 VAC.
I tried all the suggestions in the forum and only 1 works. That is putting a 10 ohm resistor across the chime terminals. Since the measured voltage was 4 volts I chose a 5W resistor just to be safe. Once install the voltage across the terminals was 1 volt so there was a 3 volt drop across the resistor. This means I could have easily used a smaller 2W resistor. Once I installed the resistor I checked device health and was still at 4000mv so all good.
Remember V*A = W. You want to choose a resistor that has a wattage rating higher than the wattage it will experience. If your voltage across the chime is different make sure you choose the correct wattage.
**This solution worked like a charm and was easy to do. ** I hope it helps some of you that are as frustrated as I was.