Magento – In the Magento Difference between Price, Group Price, Special Price, Tier Price, Display Actual Price and Manufacturer’s Suggested Retail Price

you should never call save without calling load first on a product instance.
A save without a load (on a product from a collection for example) will lead to loss of data.
If you only want to update a single attribute value for a product do this:

Mage::getSingleton('catalog/product_action')->updateAttributes(
    array($attributeId),
    array('attribute_code' => 'attribute_value'),
    $storeId
);

$storeId is the storeview for which you want to update the value. For default values use 0.

Here $product is the product Object.

Group Price:-

$roleId = Mage::getSingleton('customer/session')->getCustomerGroupId();
$grp_price = $product->getData('group_price');
$customer_groupprice = $grp_price[$roleId]['price'];

Here the current customer Group ID $roleId. $customer_groupprice is the customer Group Price.

Tier Price:-

$roleId = Mage::getSingleton('customer/session')->getCustomerGroupId();
$TierPrice = $product->getData('tier_price');
$customer_grouptier = array_column($TierPrice, 'price','cust_group');
$customer_tier = $customer_grouptier[$roleId]; 

Below the Conditions to check whether the Tier and Group Price Set or not

if(!empty($customer_groupprice))
{
echo "Customer Group Price is set";
}
else
{
echo "Customer Group Price Not Set";
}

if(!empty($customer_tier))
{
echo "Customer Tier Price is set";
}
else
{
echo "Customer Tier Price Not Set";
}