It is not possible to request only part of the page containing the div form the server, unless the contents of the div is being loaded separately on the website.
You can use XMLHttpRequest (http://developer.mozilla.org/en/docs/XMLHttpRequest) or GM_xmlhttpRequest (http://wiki.greasespot.net/GM_xmlhttpRequest) if on another domain to request the entire page, find the element in the response content, and replace the content on the page.
The exact code you need will depend a lot on the content/website, but that should get you started.