I've been seeing this same issue for around a week now. The most common (though still inconsistent / rare) way I've been able to repro it locally is by doing the following:
Open app, see that it works normally
Send new build to TestFlight
Pull to refresh in TestFlight, see that there is an update button available, but don't tap update
Try to open the app, see "Beta Has Expired" message
In this situation, if you update the app manually the app will start working as expected.
One other scenario I've only encountered once:
App updates in the background overnight to a new build in TestFlight
Try to open the app, see "Beta Has Expired" message
Open TestFlight, see that I'm on the latest version and there is no version to update to
In this situation, the fix is uninstalling and reinstalling the app.
In both situations, an automatic background update seems to be able to restore from an expired build back to a valid one. This requires submitting a new build to TestFlight and waiting, and isn't a great solution since it means we may just expose other users to the same issue and thus move the problem around without eliminating it.