How is a field's key (rather than value) obtained? In Drupal 7, I have a page content type that has a field called group (so the machine name is field_group).  The field is a List (integer) field type.

This field might have allowed values as follows:

0|Description 1
1|Description 2
2|Description 3


If I want to see the value of this field for the current page, the following code correctly works:

echo render(field_view_field('node', $node, 'field_group'));


If the key were set to 1 for this page, the above code would display the value "Description 2."

However, is there a way to instead get the key (e.g., 0, 1, 2, etc.)?
You can change the display in your content type "Manage Display" and ask to have the key and not the label :)
It's here : /admin/structure/types/manage/[your type]/display

Programmatically for a nude value as asked :

$items = field_get_items('node', $node, 'field_list'); 
$key = $items[0]['value'];