When was the last time you logged the game into Battlenet on that computer?
The game can only be played offline if it is periodically logged into Battlenet. If too much time passes, offline play is locked out until it’s logged in again.
For example, this is the screen I get when launching the game offline. The “Play Offline” button is available:
You don’t need to do anything after logging in, such as play games are or move around the UI. Just a log in and log out will suffice.