No, you don't. The App holds the route description on your iOS or Android device so as long as you download the walk on to your device when you have an internet connection then you will have the route information with you whether there is a mobile phone signal or not.
Also, the GPS tracker does not use the phone signal to pinpoint your location whether following a walk or tracking a walk. The tracker works using a satellite navigation system and therefore as long as you are in the open it will work. GPS will not work in buildings, under bridges or in caves for example. Sometimes the signal can be lost if under a dense canopy of trees or in a valley but this is rare. The tracker will also work if you carry your phone in a pocket.
However, the App does need a phone signal to download the map. If this is not available then you will still see the route track in red on your phone and a blue dot showing your location. The map background will download automatically when you have a signal.