Any site that sends X-Frame-Options headers can prevent you from being able to load the site in an iframe (in modern browsers). A related technology which could be used in some browsers is Content Security Policy which allows similar restrictions on cross-domain frames and content.
If Stumbleupon is embedding the site in an iframe, they can only support this for pages which do not serve X-Frame-Options headers with the "SAMEORIGIN" setting.
If the site is serving up the X-Frame-Options header, they're basically telling the browser, "don't embed my content in a frame." I would recommend checking this header for any site / page you'd like to embed in an iframe, and, if it's there, don't attempt including the site.
While you could work around this as you suggested by scraping the site using PHP or another tool, you're probably not following the site's wishes and could run into other problems if they detect you as a robot and start blocking your site.
X-Frame-Options MDN: X-Frame-Options
Content Security Policy HTML5Rocks: Content Security Policy MDN: Introducing Content Security Policy