I have an Obi200 (purchased late 2013 or early 2014, can't remember) that started exhibiting this problem with a new 2-line desk phone (AT&T) I installed last week. My previous 2-line desk phone didn't exhibit this problem with my Obi200, or the Obi100 before that.
It was definitely a low-level, electrical hum. And it happened only when I had 2 lines plugged in. If I only plugged in my POTS line, no hum. If I only plugged in the Obi line, no hum. And it didn't matter which jack I used for which line ... when both lines were plugged in, in any configuration, it always created the hum.
So I went and found another powersupply to use with the Obi with the same output ... changed it ... and the problem is gone. No hum. So, clearly, the power supplies that Obi is using are susceptible to this. I don't know what the replacement one came from - I've got a box of them I've collected over the years. I just grabbed the first one I found with the same output, and problem solved!