Yes, this is very important. This applies to both the iOS and the Android app. When you (accidentally) close the app nothing will sync from the CardioWatch to the app and from the app to the cloud.
You can check if the app is running by finding the Corsano app icon on your mobile device and pressing on it. If you see the Corsano logo as you start the app, it means that the app was not running.
Then, as soon as the app is running, your data start syncing. Please keep the app open. It could take some time before you see the most recent data on your phone, because the oldest data are synced first from the CardioWatch.