As far as I know, searching around the forum, there is currently no way to disable the "My Details" tab from the requester's interface.
Actually, we don't want the requester to be able to change his department for example, or the password, and we think that this option would be very useful to others too. When will this feature be present in Service Desk Plus ? We are very interested in buying this solution, but we need this option (and maybe some other minor things). And I think that the programming effort put into this shouldn't be that high.