Hi @Anonymous,
Do you want to post Outlook tasks that were completed on Microsoft Teams?
Currently, we do have a trigger for Outlook tasks, when a task is completed. However, it can be used to monitor one specific task. If you want to monitor all the tasks automatically, please take the following flow for a reference.
I would use the following flow to get tasks that were completed, and completed date right equals to Today’s date.
It is triggered by a Button. You may change it to Recurrence so that it can trigger the flow automatically daily.
Add the action List all tasks.
Add a Condition to check tasks that Status is Completed, and CompletedDate is Today. Besides, for tasks that have not been completed, the CompletedDate field would be empty. We need to use the empty function to check if the CompletedDate field is empty or not, otherwise the code will return false.
The code could be configured likes below:
@and(equals(items('Apply_to_each')?['Status'], 'Completed'),equals(if(empty(items('Apply_to_each')?['CompletedDateTime']?['DateTime']),'',formatDateTime(items('Apply_to_each')?['CompletedDateTime']?['DateTime'],'yyyy-MM-dd')),utcnow('yyyy-MM-dd')))
Under the if yes branch, add the action to Post message to Microsoft Teams.
Image for your reference.

Please take a try with it and feel free post back if you need more help.
Best regards,
Mabel