You can save your website’s URL into a variable and pass it to the shortcode:
$url = site_url();
echo do_shortcode("[timeline src="https://wordpress.stackexchange.com/questions/284281/$url/rest-of-the-url/"]");
You should also be able to use your shortcode in the following way:
echo do_shortcode("[timeline src="".site_url()."/rest-of-the-url/"]");