Someone's mangled your template code and changed how it acts when buttons are clicked.
In tpl_shopping_cart_default.php the original code reads like this:
Code:
<?php echo zen_draw_form('cart_quantity', zen_href_link(FILENAME_SHOPPING_CART, 'action=update_product', $request_type), 'post', 'id="shoppingCartForm"'); ?>
Yours has changed the 'id="shoppingCartForm"' to 'class="form-inline"' ..... which technically shouldn't break anything.
BUT yours has left the 'post' part blank, which makes the form not submit properly.