Disclaimer: I do not know how people are using continuations to build web applications, but on the ongoing debate about continuations and the web, Ian Griffith's has a small tutorial and makes the following observation which sounds sensibe:
But continuations seem to offer a solution. Whenever we get to the point in the code where we need to generate a page and feed it back to the user, we could create a continuation, store it somewhere, and then return control to the web server. When the next request comes in, a framework could retrieve that continuation, and execute it, allowing us to continue from where we left off.
You could probably cruft up a working example of this style of coding in a web app by writing an iterator function in C# which did a yield return each time it wanted to show a page to the user and wait for the results. Languages with comprehensive support for continuations could offer a more elegant example of the same idea.
However, I think this is a bad idea. Although the relationship between the code and the user navigation path is apparently simple, it hides subtle but significant details. This makes the same mistake as we did in the 1990s with distributed object models. We'd like to believe we're hiding a lot of implementation details behind a simple abstraction. In practice we're hiding important features behind an inappropriate abstraction. Our industry has more or less learned the lesson that procedure calls are an unworkably naive abstraction for request/response messaging over a network. I suspect that it's equally naive to attempt to manage the complexities of user/web server interactions with simple sequential code.
We abused iterators last year to implement our HTTP application pipeline, but am not sure this is something that can be easily generalized and is certainly not easy to explain.
In the meantime, Tomi Valkeinen has prototyped a continuations framework for Mono.
Update: Even if there is a good design for a web framework that uses continuations at its core, doesn't this require the state to be preserved in the server in some form?