Exclude certain category

Are you displaying a list of terms from a taxonomy, or are you wanting to filter a list of posts by excluding any posts who have the term in question?

For the first, displaying a list of terms from a taxonomy, it is currently not possible to filter the list in any way (unless I’ve forgotten about it, which is entirely possible 🙂 ).

If you are displaying a list of posts and wish to exclude posts who have a specific term assigned, then this is possible via PHP, but not via the shortcode. For excluding via PHP you would recite the magic words:

the_a_z_listing( array(
    'post_type' => 'your_post_type',
    'tax_query' => array( array(
        'taxonomy' => 'your-taxonomy', # this is a taxonomy slug
        'field' => 'slug', # this says that we're using slugs in the 'terms' array below
        'terms' => array( 'your-terms' ), # these are term slugs and any number of them are allowed
        'operator' => 'NOT IN',
    ) ),
) );

The code needs to be added to an appropriate template in your theme. The template-hierarchy page and image are helpful to determine an appropriate file to add or modify.