For some reason, HTC or Verizon decided to exclude FM radio from the Verizon HTC One, unlike all of its carrier variants (except for the Play Edition). Interestingly, the Broadcom FM / combo chip is the same as the one used in One Play edition. In other words, FM is disabled on it.
While on Sprint network, you can start the radio by Menu-> Settings-> Debug-> FM API to "Low Level" and restart Spirit, you cannot use that workaround with Verizon.
PS: After installing the aforementioned app, if you get stucked and are not able to turn on the Wi-Fi and Bluetooth, just force close it and everything will work fine again. This is the error you will get.
Error: No FM Accessible
SU: 2 Fm:BC2 Su:1 Bt:0 Ht:0 Mo:BCU