I think the app will not see anything if the user chooses to ignore the push notification. The workaround that comes into mind first would be to offer a way of checking the server if anything worth a push notification has happened since the last time the app was opened and correlating that with the local data.
I think the idea behind this system is to separate actual functionality from push notifications and making sure the notifications are used for only one purpose - notifying of new data/event/etc, thus leaving the downloading and processing of data to the app once it's fully in the foreground. I.e everything should be duplicated in the launching sequence to make sure the app is always up-to-date, even if the notifications during downtime were not received/ignored.
application:didReceiveRemoteNotification: is therefore meant for not transferring the data and updating the model, but reacting to the event and starting the necessary procedures. Although most of the time, the data may be so small that it fits in the notification and therefore the application can proceed without downloading any further content.