No. Once you purchase Perfect Tempo on the App Store you will not be required to pay any subscription fees or be asked to make in-app purchases to continue to use Perfect Tempo. (This does not necessarily mean that all future updates will be free - we reserve the right to charge for major new versions of the app.)
If you want to play songs from the Apple Music catalog, a subscription to Apple Music is required. This is the standard Apple Music subscription - no additional fees are required to use an Apple Music subscription with Perfect Tempo.