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);