The API docs say that an issue can be edited with a PUT request to /rest/api/2/issue/{issueIdOrKey} and only two response codes are listed: 200 and 400. However, it appears that a 204 response can also be returned, so the docs should be updated to mention that.

The request I was sending was to update a custom field. I don't know if that makes a difference, and I don't know what controls whether 200 or 204 is returned.