How to delete an eav product attribute programmatically

It is my view that this sort of thing should be done via set-up script or if that is not possible then via the admin section. I cannot think of a situation where you need to write a script for this, but maybe there is one.

For a set-up script it is incredibly simple:

$this->startSetup();
// Remove Product Attribute
$this->removeAttribute('catalog_product', 'product_attribute_code');
// Remove Customer Attribute
$this->removeAttribute('customer', 'customer_attribute_code');
$this->endSetup();

If you need something more complex like removing groups or attribute sets that can also be done via a script too.

You can try below code

Mage::getModel('catalog/product_attribute_set_api')->attributeRemove($attributeId, $attributeSetId);

It will remove attribute from attribute set only but I am not sure if it will also remove it permanently.
For removing an attribute permanently, you can try below.

Mage::getModel('catalog/product_attribute_api')->remove($attributeId);

Other method is :
First create a test.php in magento root directory
Place below code there

<?php
error_reporting(E_ALL | E_STRICT);

require_once './app/Mage.php';
umask(0);
Mage::app();

$attr = 'test_remove'; //attribute code to remove

$setup = Mage::getResourceModel('catalog/setup', 'core_setup');
try {
    $setup->startSetup();
    $setup->removeAttribute('catalog_product', $attr);
    $setup->endSetup();
    echo $attr . " attribute is removed";
} catch (Mage_Core_Exception $e) {
    print_r($e->getMessage());
}