WP_Query for custom post type and category_id not working

category__in is used for built-in post categories, not for custom taxonomies.

You can try something like this:

$args = array(
    'post_type' => 'tribe_events',
    'meta_query' => array(
        array(
            'key'     => '_EventStartDate',
            'value'   => array( $start_date, $end_date ),
            'compare' => 'BETWEEN',
            'type'    => 'DATE'
        )
    ),
    'tax_query' => array(
        array(
            'taxonomy' => 'tribe_events_cat',
            'field'    => 'term_id',
            'terms'    => array( 2, 6 )
        )
    )
);

$query = new WP_Query($args);