No, your mobile device does not need to connect to internet for it to work and connect your Sensor. The OvuSense App will work offline for up to 30 days and you will receive a warning up to that point that you need to connect to the internet.
Beyond day 30 all your previous data will be retained by the OvuSense App but you will no longer be able to connect your OvuSense Sensor until you have connected to the internet again.