Code remove a product in cart.
$cartHelper = Mage::helper('checkout/cart');
$items = $cartHelper->getCart()->getItems();
$sku = 'product_1';
foreach ($items as $item) {
//remove product have sku is $sku
if ($item->getProduct()->getSku() == $sku) {
$cartHelper->getCart()->removeItem($item->getItemId())->save();
}
}
