For the calculation of the fare: NO. You can perfectly start a trip and complete the whole trip process till the end without Internet connection.
For backend services, like upload of trip data to the server: YES, but if there is no Internet connection, app will try to connect regularly.
A driver login and logout needs Internet access, but in case there is none, the driver can login and logout locally on the device.
For the calculation of a quote, Internet is needed.