If the user does not have the appropriate permissions (in our case somebody referred to a read only issue), it crashes and will try again for the next build (it will post some more comments into the tickets that the user has permissions to and once it reaches the read-only issue ID, it will crash again, and so on)

The problem lies here:
https://github.com/jenkinsci/jira-plugin/blob/master/src/main/java/hudson/plugins/jira/Updater.java#L138

According to the JIRA documentation, it should catch a RemoteException instead:

com.atlassian.jira.rpc.soap.addComment

Throws:

• RemotePermissionException - If the user is not permitted to perform this operation in this context.
• RemoteAuthenticationException - If the token is invalid or the SOAP session has timed out
• RemoteException - If the issue does not exist or your dont have permission to see it

Source:
http://docs.atlassian.com/rpc-jira-plugin/latest/com/atlassian/jira/rpc/soap/JiraSoapService.html#addComment(java.lang.String, java.lang.String, com.atlassian.jira.rpc.soap.beans.RemoteComment)