<p>I have heard of desktop applications whose code has been open-sourced, but are there any websites whose code has been open-sourced? I don't think I have heard of any. If there are aren't any, are there any reasons why?</p>

<p>This will have the advantage of not having to wait for the web development team to include a new feature. Also, this would ensure that more eyes see the code, ensuring a lower bug count. Of course, hacking would be easy.</p>

Pastebin:
http://pastebin.com/
The wiki behind Wikipedia et al:
http://www.mediawiki.org/wiki/MediaWiki
GrailsCrowd:
http://grailscrowd.com/
Slashcode [slashdot engine]:
http://www.slashcode.com/
The reason that not as many people do it is because it would be easy for someone to set up a complete clone of your website without making any useful changes. OK, that's strictly allowed with true 'open source' code, but it's not quite within the spirit of the thing (IMO) and if the site became even moderately popular there would be a
lot
of useless clones. The site would become essentially worthless.
Those are -websites- that are open source. There is a huge amount of web application software that is open source [some of it shitty]
WordPress
InkType
Joomla
osCommerce
ZenCart
Drupal