No! One of the many benefits of PulseOn is that it works independently, storing all your training data onto the device memory and automatically syncs the details into the application when you are close to your mobile device the next time. Thus there's no obligatory carrying your phone with you!
For heart rate measurement, calories, training effect and training intensity the phone is not needed. The device will transfer training events to the application when synchronized again. The memory of the PulseOn device is enough for tens of hours of exercise.
However, to estimate your fitness level the application will need to know your speed and distance while you are running or walking outdoors, so you should use your phone every now and then while exercising to keep your fitness level up to date.