To delete an attachment on an Action Item, Navigate to Action Items -> All Action Items or My Action Items and find the Action Item where you need to delete the attachment.
Click on the black arrow (highlighted below) of the Action Item to have the messages tab drop down and then click on the green paper clip to the right to view attachment.
A separate Attachments window will appear. Click the arrow button on the left (highlighted) and then the delete button appears.
If there are more than one attachments, you can switch between different attachments on the action item by pressing the Prev and Next buttons.
Select Delete button and pop up message will appear. Select OK button to Delete.