Does the RescueTime app have to be running to collect the data?

The short answer: Yes, the RescueTime app must be running in order to collect data.

By default the RescueTime application will collect the data and send that data to the RescueTime.com servers every 30 minutes. However, if you happen to not have a internet connection, RescueTime will queue up the collected data and send it the next time you are connected, so you can use RescueTime even when off-line.

Hope that helps, let me know if you have any other questions.