I also started having this problem about 2 days ago, and it started after upgrading to iOS 7.1 last week. The phone keeps switching in and out of silent mode (without the button being touched) and mostly stays in silent.
I tried restarting, reseting, restoring, cleaning the headphone jack, and nothing helps. I thought maybe it was stuck due to a headphone issue and I had minor success with a strage combination of restarting, switching out of mute and unplugging the headphones, but it lasted about 5 minutes and went back into silent.