Wordpress - Posts with at least 3 tags of a list of tags

The answer below is simplified, and could be extended to check if any posts have 3 matching tags before outputting the list. Using one query and assuming you have at least one post with 3 matching tags:

//List of tag slugs
$tags = array('foo', 'bar', 'chocolate', 'mango', 'hammock', 'leaf');

$args = array(
    'tag_slug__in' => $tags
    //Add other arguments here
);

// This query contains posts with at least one matching tag
$tagged_posts = new WP_Query($args);

echo '<ul>';
while ( $tagged_posts->have_posts() ) : $tagged_posts->the_post();
   // Check each single post for up to 3 matching tags and output <li>
   $tag_count = 0;
   $tag_min_match = 3;
   foreach ( $tags as $tag ) {
      if ( has_tag( $tag ) && $tag_count < $tag_min_match ) {
         $tag_count ++;
      }
   }
   if ($tag_count == $tag_min_match) {
      //Echo list style here
      echo '<li><a href="'. get_permalink() .'" title="'. get_the_title() .'">'. get_the_title() .'</a></li>';
   }
endwhile;
wp_reset_query();
echo '</ul>';

EDIT: Adjusting the variable $tag_min_match will set the number of matches.

Here's one way to do it:

Given a set of 5 tags, {a, b, c, d, e}:

1) In PHP, generate all the possible subsets containing 3 elements, without repetition:

{a, b, c}
{a, b, d}
{a, b, e}
{a, c, d}
{a, c, e}
{b, c, d}
{b, c, e}
{c, d, e}

2) Convert those subsets into a massive taxonomy query:

$q = new WP_Query( array(
  'tax_query' => array(
    'relation' => 'OR',
    array(
      'terms' => array( 'a', 'b', 'c' ),
      'field' => 'slug',
      'operator' => 'AND'
    ),
    array(
      'terms' => array( 'a', 'b', 'd' ),
      'field' => 'slug',
      'operator' => 'AND'
    ),
    ...
  )
) );