link_to generates a <a> tag, which is not input type="submit". What you should use should be a button_to, which generates a form with a input type="submit" button to the link:
<%= button_to "Add", new_admin_course_path, :id => "open-contacts-dialog-btn",
:class => "inbox-sf-add-btn tip", :method => :get %>
Note the :method => :get. Without it, the button_to will generate a form with method set to post.