if it's actual expiration of a time based licence (ie, student, demo, or some other thing) you should check what the restrictions are for that licence.
I don't think there's any restrictions on doing open source designs and releasing them for free, but your own licence will be the ultimate source of truth there (the demo and student licences do forbid you doing commercial work of any type while you use them, for instance)
but whatever happens with the licence, once it's expired, the files you generated with it are still good for anyone else with a valid altium licence to pick up and use... the project files themselves aren't tied to the licence that created them, and don't expire.