The canonical continuation-based web framework for Haskell is the venerable WASH system.
The idea is to capture state in the continuation, allowing for fully RESTful, stateless web apps that can in some cases be generated automatically from the non-continuation-based version of the program.
From "WASH/CGI: Server-side Web Scripting with Sessions and Typed, Compositional Forms": (2001):
The main idea is to use a continuation to take a snapshot of the state
of the script after sending the form to the browser. This continuation
is then stored on the server and the form contains a key for later
retrieval of the continuation.
A modern description of the approach is given in the MFlow Monad Reader overview.