The Navigation bar on which the Refresh button resides is a core security feature. There are replacement Toolbars for the Navigation bar (quero.at) on which you can choose to include the Refresh button or not, but unfortunately this solution would not be available to the general public visiting your site.
You could use Modal or Modeless windows that have the toolbar option switched off or if your web site is on a publicly available network choose to run IE only in kiosk mode. All of these methods can provide a browser interface that does not display the Navigation bar by default.
see. showModalDialog, showModelessDialog and createPopup
I have not tested it but from memory I think the refresh button only refreshes from the cache, so if you allow your web pages to be cached then in effect pressing the refresh button will not send a request back to your server, then you only have to worry about disabling the Ctrl+F5 shortcut that forces a refresh from the server.