This behavior is usually the result of not being in an administrator account when you try to authorize. Regular and limited users do not have the necessary permissions for the authorization to hold after you close the program.
For authorization to hold, you must log in to an administrator account on that computer and perform the authorization. This authorization will also unlock the program for all other user accounts on this computer.
On Windows Vista, Windows 7, Windows 8, and Windows 10, you will need to open the program by right-clicking the Desktop icon for the application and selecting Run as administrator in order to authorize properly.
If you are using Finale (Version 25) on Windows 10...
There is a known issue that has been reported for some users of Finale v25 on Windows 10 where an authorization will not stick, and you eventually run out of authorizations. In order to rectify this, make sure that you are on the latest version of Finale. This issue and others related to it were resolved in Finale v25.4.