We have written the Auto-Close Answered Tickets functionality into a new branch of osTicket and submitted a Merge Request here https://github.com/osTicket/osTicket/pull/6051
This feature now allows you to automatically close answered tickets after the required number of days and can be set per department.
We have also created a new email template which you can also enable or disable per department that sends an auto response when a ticket is closed.
The new feature will be in the Admin Panel > Agents > Departments > [Department Name] under the Autoresponder Settings section.
The mail template for the email can be found in Admin Panel > Emails > Templates The template is in the Ticket End-User Email Templates section and is called Closed Ticket Auto-reply
Anyone interested in getting this merged into the core of osTicket please can you comment on this in the Pull Request which can be found here - https://github.com/osTicket/osTicket/pull/6051
If can't wait for the changes to be merged or if the pull request does not get enough momentum you can download it here - https://github.com/datatecuk/osTicket/tree/Auto-Close-Answered-Tickets
Hopefully with enough people asking for it over the years and responding on the Pull Request the project developers will merge it into the core.