Magento 2 – Get Quote Item Collection by Quote Item ID

Don't do this.

Your problem is, that the cart is not yet saved, have a look here:

https://github.com/LokeyCoding/magento-mirror/blob/magento-1.7/app/code/core/Mage/Checkout/controllers/CartController.php#L201-L206

public function addAction()
{
// ...
        $cart->addProduct($product, $params); // <-- you are inside this method
        if (!empty($related)) {
            $cart->addProductsByIds(explode(',', $related));
        }

        $cart->save(); // here is the saving, and therefore after this line,
                       //  quote and items have an id
// ...
        Mage::dispatchEvent('checkout_cart_add_product_complete',
            array('product' => $product, 'request' => $this->getRequest(), 'response' => $this->getResponse())
        );

What you want is to listen on checkout_cart_add_product_complete

If you want to know which items where added this round, just flag them in checkout_cart_product_add_after like $quoteItem->setIsNew() then you can check in checkout_cart_add_product_complete for $quoteItem->getIsNew()

try below code because save method doesnot work with collection

$quote = Mage::getModel('sales/quote')->load(quote_id);
// get item id by your logic 
$quote->removeItem($itemId);
$quote->save();

removeItem method code

public function removeItem($itemId)
{
    $item = $this->getItemById($itemId);

    if ($item) {
        $item->setQuote($this);
        /**
         * If we remove item from quote - we can't use multishipping mode
         */
        $this->setIsMultiShipping(false);
        $item->isDeleted(true);
        if ($item->getHasChildren()) {
            foreach ($item->getChildren() as $child) {
                $child->isDeleted(true);
            }
        }

        $parent = $item->getParentItem();
        if ($parent) {
            $parent->isDeleted(true);
        }

        Mage::dispatchEvent('sales_quote_remove_item', array('quote_item' => $item));
    }

    return $this;
}