I had the same problem, two list boxes and two buttons.
The data in the list boxes was being loaded from a database and you could move items between boxes by clicking the buttons.
I was getting an invalid postback.
turns out that it was the data had carriage return line feeds in it which you cannot see when displayed in the list box.
worked fine in every browser except IE 10 and IE 11.
Remove the carriage return line feeds and all works fine.