Drupal - Programmatically delete one entry from one field on one entity

If you're able to rely on the entity API module you should be able to use code similar to the following:

// Load some entity.
$entity = entity_load_single($entity_type, $id);

// Remove the field value.
unset($entity->field_FIELD_NAME[LANGUAGE_NONE][$index]);

// Reset the array to zero-based sequential keys.
$entity->field_FIELD_NAME[LANGUAGE_NONE] = array_values($entity->field_FIELD_NAME[LANGUAGE_NONE]);

// Save the entity.
entity_save($entity_type, $entity);