What happens to a task's due date when I push it?
The date moves with the task into your task app, and then it's removed from your Supernote so it stops cluttering the device's Planned view. The app still shows the original date in its own "Pushed tasks" folder, so you don't lose sight of it. (Pulling an item back out of the Pushed list on the device currently doesn't restore the date to the device.)