The user in this topic had a similar issue: https://www.lansweeper.com/forum/yaf_postst15999_Ticket-Closed-State-Change-Email-Not-Working.aspx#post53872
Note that if you send a public note and change the state at the same time, it will not send the outgoing email template.
If you are using the other methods to close the ticket. Check your mailbox sent folder to see if the outgoing email template was sent from your mailbox or not.
You can also do tests with the MailTester found under Program Files (x86)\Lansweeper\Service to test general email functionality. But I would assume this will not return any unexpected results.
Lastly, if you are not yet on the latest version of Lansweeper, I'd recommend updating. Instructions can be found here: https://www.lansweeper.com/kb/118/updating-your-installation.html