Select event where start date is smaller than current date and end date is greater than current date

You can try to replace this kind of comparison:

AND m1.meta_value >= '$start_date'

with:

AND CAST(m1.meta_value AS DATE) >= '" . date( "Y-m-d H:i:s", $start_date ) . "'

or if you would use Unix timestamp for the meta values then you could try:

AND m1.meta_value+0  >= ". strtotime( $start_date ) ."

Update:

Here is a modification of your query:

$query = " SELECT DISTINCT * FROM {$wpdb->posts}
INNER JOIN {$wpdb->postmeta} m1 ON ( {$wpdb->posts}.ID = m1.post_id )
INNER JOIN {$wpdb->postmeta} m2 ON ( {$wpdb->posts}.ID = m2.post_id )
WHERE {$wpdb->posts}.post_type="tribe_events"
AND {$wpdb->posts}.post_status="publish"
AND (
       ( ( m1.meta_key = '_EventStartDate' AND CAST( m1.meta_value AS DATE ) >= %s ) AND ( m2.meta_key = '_EventStartDate' AND CAST( m2.meta_value AS DATE ) <= %s ))
       OR 
       ( ( m1.meta_key = '_EventEndDate' AND CAST( m1.meta_value AS DATE ) >= %s ) AND ( m2.meta_key = '_EventEndDate' AND CAST( m2.meta_value AS DATE ) <= %s ) )
       OR
       ( ( m1.meta_key = '_EventStartDate' AND CAST( m1.meta_value AS DATE ) < %s ) AND ( m2.meta_key = '_EventEndDate' AND CAST( m2.meta_value AS DATE ) > %s ) )
    )
ORDER BY {$wpdb->posts}.post_date DESC;";

$query = $wpdb->prepare( $query, $start_date, $end_date, $start_date, $end_date, $start_date, $end_date );
$result = $wpdb->get_results( $query );

where the two INNER JOINS are added since you are filtering with AND on two different meta keys.

Update:

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