This is a pretty common support request, and the gist of it is you cannot use Views form functionality on Views embedded in the checkout form. The reason is pretty simple to understand - you end up nesting a FORM tag within a FORM tag because Views doesn't know that it's building form elements to render inside another form. The solution is not so simple.
The best advice I've been able to give thus far is to disable the cart contents checkout pane and create a Views block instead that uses the form functionality. However, if someone removes an item from the cart after making a change elsewhere on the page, the page would refresh and that data would be lost. Thus you'll want to figure out a way to have that remove button work via AJAX, which is totally possible through the forms API but remains undocumented as far as I know. If you do code a solution, a blog post about it or doc page on d.o would be great.