The only purpose of a cheeseburger without cheese that I can figure out has come to me through a lunch experience with a friend. Time and time again he will order the same thing and it took me a while to figure it out.
He would order a Number 2 value meal at McDonald's, which is two cheeseburgers, fries, and a drink. But he would ask for them without cheese. So instead of a two-cheeseburger-meal, he would get a two-hamburger-meal, which still ended up being cheaper than if you bought the items separately.
On a side note, he would also get the 99 cent double cheeseburgers without cheese.