You would most likely need to use a function in your theme (child theme I hope, otherwise make your theme into a child theme – much safer) functions.php file.

You need the ID number of the search form as well and the class name you want to remove. So, here is some sample code:

global $searchandfilter;
add_filter( 'body_class','remove_my_body_classes' );
function remove_my_body_classes( $classes ) {
  if ( $searchandfilter->active_sfid() == 35889) {
    if(($key = array_search('class_name_here', $classes)) !== false) {
      unset($classes[$key]);
				}
    return $classes;
  }
}

Replace the number 35889 with your search form ID number (from the shortcode) and replace the phrase body_class with whatever you need to remove.