From their website:
FiOS Service
Verizon FiOS Internet Service uses fiber optic technology to deliver phone and broadband Internet service to your home.
This fiber optic technology uses strands of optically pure glass that carry digital information as pulses of light. With FiOS, hundreds of thousands of these strands are bundled together to make the fiber optical cable that is connected to your home.
When you access your phone or the Internet, a fiber optic cable carries the laser-generated pulses of light to transmit the data to and from your home. At your home, an Optical Network Terminal (ONT) converts it to an electrical signal in the appropriate way.
Therefore, it appears it MAY Be connected to your phone jacks. Therefore, you could NOT connect an OBi to your phone jacks.
I would think that you could easily see how your phones are hooked up. If they are plugged into the wall, there's your answer. You would have to isolate which wires FIOS is hooked to, and then use another pair at the outlets you want connected to the OBi. Or, connect a wireless phone base to the OBi and put the remotes through the house.