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.
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.)