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?
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.
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]
Check out reddit.com's source code here: http://code.reddit.com/. I do think there is a profit consideration when a site is built, for them to release their source code would allow copy-cat sites and dilute their market share.
PHP.net is open-source. Hell you can checkout a copy of their website if you want to.
SugarCRM OSCommerce DotNetNuke Mambo
There's quite a bit of software that are open source.
Wikipedia/Mediawiki is open source on multiple levels. MediaWiki (the software which runs WikiPedia) is open source, and accepts contribumtions, and the content of WikiPedia is also open source and editable by everyone.
There's also LiveJournal (http://www.livejournal.com) which has been open source for years.
Many answers have addressed the competitive aspect, but that really applies equally to all FOSS projects, web-based or not. The distinctive feature of web sites is that, by and large, the users of the software do not have access to the code in any form - source or binary - so the typical FOSS requirement that you make the source available to anyone who receives the binary becomes pretty much meaningless. I believe this to be the primary impediment to widespread use of FOSS licenses in conjunction with websites.
(There have been recent attempts to develop web-focused FOSS licenses which specifically address this issue, but none have really caught on yet.)