In some rare cases where you would like to be able to close a Request without notifying the customer, please follow the below mentioned:
1. Go to Admin > Notification Rules > Choose the desired Account and disable the rule called, "Acknowledge requester by e-mail when the request is closed.".
2. Copy the content of the above notification rule.
3. Go to Custom Triggers > Email Notification and paste the above content by setting the "TO" as $Requesters and set the criteria's as :
# When a Request is "Created & Edited"
# Requester Name is not --"select the user's name whom you don't want to send notification".
# Status is Closed
Sharing screenshot for reference: