Wordpress - Redirect function inside a Shortcode

As @Rarst explains, shortcodes normally run too late for you to redirect from inside one. They usually run on the the_content hook which is well after content is sent to the browser. If you need to redirect based on the presence of a shortcode you need to check for that shortcode before any content leaves the server.

function pre_process_shortcode() {
  if (!is_singular()) return;
  global $post;
  if (!empty($post->post_content)) {
    $regex = get_shortcode_regex();
    preg_match_all('/'.$regex.'/',$post->post_content,$matches);
    if (!empty($matches[2]) && in_array('yourshortcodeslug',$matches[2]) && is_user_logged_in()) {
      // redirect to third party site
    } else {
      // login form or redirect to login page
    }
  }
}
add_action('template_redirect','pre_process_shortcode',1);

That is "proof of concept". The particular conditions you need will likely be different. Note that that is a pretty "heavy" bit of processing. I would make sure it only runs where absolutely necessary.

wp_redirect() performs redirect via HTTP headers so technically it won't (or at least shouldn't) work after page output started. So you can't just use this function in shortcode.

Which doesn't mean however you can't use shortcode to control it. You could check for fitting conditions (if is page and page contains shortcode) before output started (somewhere around template_redirect hook) and perform redirect then.

Another option would be to output conditionally JavaScript that will perform redirect after page have loaded.