Short version:

$ export EDITOR='emacsclient -c'
$ sudoedit /etc/foo.conf

Of course emacsclient can't find the Emacs server when you run it as
root--root isn't running an Emacs server.

Furthermore, if root's emacsclient *did* connect to the normal user's Emacs
server, you would have the normal user's permissions and still wouldn't be
able to edit the file.  ;-)

-- 

-PJ