pre get posts changing the query

Try using meta_query parameter :

add_action('pre_get_posts', 'add_event_date_criteria');
function add_event_date_criteria(&$query)
{
    // We only want to filter on "public" pages
    // You can add other selections, like here:
    //  - Only on a term page for our custom taxonomy
    if (!is_admin() &&
        is_tax('event-tag') || is_tax('event-category')) {
        $time = time();
        $meta = array(
            array(
            'key' => 'start_time',
            'value' => $time,
            'compare' => '>='
            )
        );
        $query->set('meta_query',$meta );
        $query->set('meta_key', 'start_time');
        $query->set('orderby', 'meta_value_num');
        $query->set('order', 'ASC');
    }
}

also you need to make sure your meta field start_time is measured and saved in the number of seconds since the Unix Epoch like time() function.