I've seen this question while researching the issue myself, but could not find a solution to it.
I did however see that you can change the title of the position tile from open to retired by doing the following:
First, search for and open the position that you've retired.

Then click on the Changes timeline button at the top of the ribbon, choosing to open the manage changes button:

Select the tab Position hierarchies and remove/delete the record(s) here.

Click on the Save button on the top left of the same window to ensure the removal has been submitted.
Then, even if you have retired the position with a retirement date, make sure that there is a record that holds that retirement date against the position record. To do this, click on the tab Position durations on the same window to see if there is a record. If there isn't add one, making sure that the activation and retirement date is in the past.

This I found out on an answer from André who stated that the hierarchy works within valid date ranges. Reference:
https://community.dynamics.com/ax/f/microsoft-dynamics-ax-forum/110661/position-issue/221159#221159
Heading back to the position hierarchy view, it will be NOT be removed from the view, but show the position as retired instead of open.
How can I remove the position completely from the hierarchy? Deleting the position is the only way I can see at the moment, but even then, I cannot do this...
Please help