How do I exclude the current post from the upcoming post query

I think you are asking how to exclude the current exhibition from your upcoming exhibitions query.

If that is the case, simply insert the ID of the current exhibition into the query using ‘post__not_in’:

$exhibition_upcoming_query = array(
  'post_type' => 'exhibition',
  'orderby' => 'meta_value',
  'order' => 'ASC',
  'posts_per_page' => 3,
  'post__not_in' => array( $exhibition_current_loop->posts[0]->ID ),
  'meta_query' => array(
          'key' => 'exhibition_start_date',
          'value' => date('Y-m-d'),
          'compare' => '>',
          'type' => 'DATE',

Hit me up if I am misunderstanding what you are after.