Route planning:
To plan a route in the app your phone needs to have an active internet connection, either via mobile data or WiFi. Once a ride has been planned and saved in the Beeline app, you can select and ride it without needing an internet connection.
To import a GPX file in Route Mode you will also need an internet connection.
However, you can plan a ride or import a GPX file in Compass Mode without an internet connection, and start navigating as usual. This is because Compass Mode only uses GPS info and doesn't need to plot a new route.
Navigating:
When navigating a pre-planned journey or saved route, Velo will work offline.
That means that if you lose signal or want to save phone battery by using Airplane Mode, the Velo will keep navigating you correctly along the route that you planned.
One thing to bear in mind, though, is that your phone will need mobile data for automatic rerouting, if you go off-course. If you have mobile data turned off, it will only be able to point you back to where you left the planned route.
The only feature that will not work offline/without a data connection is re-routing.