Apache Solr: get list of "More Like This" nodes programmatically
Here is the solution that I came up with.

function mymodule_similar_nodes($node) {
  if (module_exists('apachesolr_search')) {
    $output = '';
    $env_id = 'solr'; // machine name of solr server
    $delta = 'mlt-002'; // "More Like This" block delta
    $block = apachesolr_search_mlt_block_load($delta);
    $solr = apachesolr_get_solr($env_id);
    $context['search_type'] = 'apachesolr_search_mlt';
    $context['block_id'] = $delta;
    $docs = apachesolr_search_mlt_suggestions($block, apachesolr_document_id($node-&gt;nid), $solr, $context);
    $view_mode = 'teaser'; // view mode, e.g. 'full', 'teaser'...
    foreach ($docs as $doc) {
      if ($doc-&gt;entity_type == 'node') {
        $similar_node = node_load($doc-&gt;entity_id);
        $similar_node = node_view($similar_node, $view_mode);
        $output .= drupal_render($similar_node);
      }
    }
    return $output;
  }
}