If you follow Obihai's directions for putting GV onto your Obi box and setting up your Google Chat a.k.a. Hangouts, then your GV number will ring the phone plugged into your Obi box. When you pick up that phone to make a call, your caller ID will be your GV number.
You won't need to stay logged into GV or Hangouts on your computer, if that is your question. If you discontinue your Google Voice membership, though, that probably will stop your usage of Obi with your GV number.