Multiple Queries with meta_query

You could try this which is mostly code clean up but may solve your problem. I removed the relation from meta_query as it’s only needed if you have multiple meta keys in your query, which you don’t. I’ve also replaced the rewind_posts() with wp_reset_postdata() and finally, you were ending your IF statements earlier than your UL tags, which if it is false you’ll have orphan </ul>‘s hanging around which is never good.

<h1>2013-14 Concert Season</h1>
    <?php
        $today = date('Ymd');

        $args1 = array(
            'type' => 'post',
            'category_slug' => 'Featured1314',
            'meta_key' => 'concert_date',
            'meta_query' => array(
                    array(
                        'key' => 'concert_date',
                        'value' => $today,
                        'compare' => '>='
                    )
                ),
            'orderby' => 'meta_value_num',
            'order' => 'ASC'
            );
        $upcoming_query = new WP_Query($args1); 

        if( $upcoming_query->have_posts() ) { ?>
            <h2>Upcoming Concerts</h2>
            <ul><?php while ( $upcoming_query->have_posts() ) { $upcoming_query->the_post(); ?>
                <li id="post-<?php the_ID(); ?>"><h1><a href="https://wordpress.stackexchange.com/questions/149415/<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></h1><p><?php the_excerpt() ?></p></li>
            <?php } // endwhile ?>
            </ul>
        <?php } // endif;

        wp_reset_postdata();
        $args2 = array(
            'type' => 'post',
            'category_slug' => 'Featured1314',
            'meta_key' => 'concert_date',
            'meta_query' => array(
                    array(
                        'key' => 'concert_date',
                        'value' => $today,
                        'compare' => '<'
                    )
                ),
            'orderby' => 'meta_value_num',
            'order' => 'ASC'
        );
        $past_query = new WP_Query($args2); 
        if( $past_query->have_posts() ) {?>
            <h2>Past Concerts</h2>
            <ul><?php while ( $past_query->have_posts() ) { $past_query->the_post(); ?>
                <li id="post-<?php the_ID(); ?>"><h1><a href="https://wordpress.stackexchange.com/questions/149415/<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></h1><p><?php the_excerpt() ?></p></li>
            <?php } //endwhile; ?>
            </ul>
        <?php } // endif;