It is possible to search for posts with a specific term by appending the URL with:

?&term=abc

Is it possible to do the opposite and search excluding a specific term? Something like:

?&term=NOTabc

The basic explanation

1. You have a template tag that is called is_search() to determin if you’re on a search page or not.
2. This then calls get_search_template() which basically is a wrapper function for get_query_template('search').
3. When you look into the last function, then you’ll see that it basically does locate_template(), which checks for file existence and then does a load_template().
4. There your $wp_query->query_vars get checked and the $_template_file gets extracted and loaded.

A note before about the query var …

WordPress uses the query var s to save the searched term into its query. If you’re running a custom query string, then you need to modify the get_search_query filter like the following:

/**
 * Modify search query var
 * Doesn't need to be wrapped into esc_attr(), as it's already done by core
 * 
 * @param string $s | The query var to save the search in
 * @return string $s
 */
function change_search_query_var( $s )
{
    $s="your_custom_query_var";

    return $s;
}
add_filter( 'get_search_query', 'change_search_query_var', 20, 1 );

Some solution(s)

We now have different possible solutions:

The easy solution

^{Runs in the template and better fits for excluding taxonomy terms.}

A) Get all posts in the template loop, check every post in the loop for every (taxonomy) term ¹⁾ and then exclude/don’t show it.

In you loop in your template search.php file:

if ( have_posts() ) {
    while ( have_posts() ) {
        the_post();

        global $post;
        // Skip if the post has the term attached
        if ( is_object_in_taxonomy( 
            $post->ID,
            'YOUR_TAXONOMY',
            array( 'TERM A', 'TERM B', '...' ) 
        )
        continue;
    } 
}

This is just an example of the loop.

The sophisticated solution(s)

^{Runs before running the actual main query and suits better for excluding terms in general.}

B.1) Filter the term out of of the search template query.

In the following example I use the posts_clauses filter to show you that you can modify even more with just one filter (do a var_dump of the $pieces array for more insights). You could also use the posts_where filter, which runs before the clauses filter.

// In your functions.php file
/**
 * Modify the search query where clause
 * Like escapes the term for security reasons
 * 
 * @param array $pieces | The array of post clauses: Where, Group by, etc.
 * @return array $pieces
 */
add_filter( 'posts_clauses', 'alter_search_query', 10, 2 );
function alter_search_query( $pieces, $query )
{
    // Target all search queries in the front-end:
    if( is_admin() || ! $query->is_search() ) return $pieces;

    global $wpdb;
    $term = $wpdb->esc_like( 'YOUR_TERM' );

    $pieces['where'] .= $wpdb->prepare( 
        " AND   {$wpdb->posts}.post_title NOT LIKE '%s'",
        "%{$term}%" 
    );

    return $pieces;
}

B.2) Filter the term out of of the search template query _in a more performant & specific way.

In the following example I use the posts_search filter to show you how you can modify the where clause only for the search query. It’s pretty much the same as the posts_where filter.

// In your functions.php file
/**
 * Modify the search query where clause
 * Like escapes the term for security reasons
 * 
 * @param array $pieces | The array of post clauses: Where, Group by, etc.
 * @return array $pieces
 */
add_filter( 'posts_search', 'alter_search_where', 10, 2 );
function alter_search_where( $search_term, $query )
{
    // Target all search queries in the front-end:
    if( is_admin() || ! $query->is_search() ) return $search_term;

    global $wpdb;
    $term = $wpdb->esc_like( 'YOUR_TERM' );

    $search_term .= $wpdb->prepare( 
        " AND   {$wpdb->posts}.post_title NOT LIKE '%s'",
        "%{$term}%" 
    );

    return $search_term;
}

Footnotes:

^{1) You haven’t been clear if it’s about a search or a taxonomy term.}