Wordpress - How to use wp_nav_menu to create a select menu dropdown?

You can't do this with wp_nav_menu, because it outputs list items, and you'll generate invalid markup with your code.

Try using wp_get_nav_menu_items() instead.

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A quick solution for a drop down menu with a custom walker:

class Walker_Nav_Menu_Dropdown extends Walker_Nav_Menu{

    // don't output children opening tag (`<ul>`)
    public function start_lvl(&$output, $depth){}

    // don't output children closing tag    
    public function end_lvl(&$output, $depth){}

    public function start_el(&$output, $item, $depth, $args){

      // add spacing to the title based on the current depth
      $item->title = str_repeat("&nbsp;", $depth * 4) . $item->title;

      // call the prototype and replace the <li> tag
      // from the generated markup...
      parent::start_el(&$output, $item, $depth, $args);
      $output = str_replace('<li', '<option', $output);
    }

    // replace closing </li> with the closing option tag
    public function end_el(&$output, $item, $depth){
      $output .= "</option>\n";
    }
}

In your templates use it like this:

wp_nav_menu(array(
  'theme_location' => 'primary', // your theme location here
  'walker'         => new Walker_Nav_Menu_Dropdown(),
  'items_wrap'     => '<select>%3$s</select>',
));