What happens once your total storage space has been filled?

Does it simply automatically delete submissions chronologically starting with the oldest submission, retaining the most recent?

Also does this interrupt any current forms?

For submission storage, yes, the system automatically deletes submissions chronologically starting with the oldest submission, retaining the most recent 500.  We keep those submissions that are delete for 90 days before fully deleting them, though they are removed from the account before that time.  Being over the submissions storage limit will not affect the functionality of your forms.

Upgrading to Storage or above will restore those submissions.