The weight is displayed by this template file:
skins/default/en/shopping_cart/parts/item.info.weight.tpl
If you check the file, it has the @ListChild directive that says that the template must be shown in the "cart.item.info" list. So, to removing the template from that list should be enough to hide the weight in the shopping cart.
I haven't tested the following code, but creating a custom module and adding the following method to your Main.php (or doing the same for Custom Skin module) should work:
PHP Code:
protected static function moveTemplatesInLists()
{
return array(
'shopping_cart/parts/item.info.weight.tpl' => array(
static::TO_DELETE => array(
array('cart.item.info', \XLite\Model\ViewList::INTERFACE_CUSTOMER)
),
),
);
}