I figured this one out by accident as I was migrating websites to a new server. When the last website not assigned to a subscriber was moved off the server the "Install Help Desk Link" disappeared.
So if you have any domains in the "My Webspaces" on the panel home page (i.e. not assigned to a subscriber) then you will see the "Install Help Desk Link". If you create a new customer and assign those domains to this new customer then the link will disappear.
Not sure what the rationale is for this behaviour but that seems to be the way it works.