No; Coin works as a stand-alone device and can be used independent of your mobile device.
Think of your mobile device as a companion to your Coin that updates it with information. Your mobile device is needed for the initial set-up, for adding or changing cards on your Coin, and to fully utilize Security features. For these reasons, we recommend keeping your mobile device with you. Nonetheless, you can always use your Coin without your mobile device - like in case it runs out of battery, you leave it at home, or it's in Airplane mode.