Looking at the issue triggers, there are options for creating, releasing, unreleasing, and updating releases (see image below). However, I don't see anything about archiving releases.
Is there another way to automatically archive a release through Jira Automation?
At this time, archiving from a rule with an action is not possible. That is in the backlog:
As an elaborate work-around, you could call the REST API to do this by sending a web request action. Here is some how-to information if you want to try that: