How to turn off "Notifying customers of case status changes"

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15 
$hook_array['before_save'][] = Array(10, 'Save case updates', 'modules/AOP_Case_Updates/CaseUpdatesHook.php','CaseUpdatesHook', 'saveUpdate');

This points to the function which sends the email. To adjust it to not send an email you would need to create a class CustomCaseUpdatesHook in custom/modules/AOP_Case_Updates/CaseUpdatesHook.php which extends the CaseUpdatesHook class, and write a new function for sendCaseUpdate.