If you were propagating e.g. game state, you'd still want to stick that in a db, so the continuation would just be a session id?
Game state has to be propagated from the server to the client so the player knows what is happening where-as continuations as used in this article are more about avoiding this propagation (the hidden field in the example is replaced by dispatch-table tag which acts as a session id / location in the dispatch-table for finding the continuation function)
If you google ‘serializable continuations’ you’ll find a reasonable amount of prior art (e.g. http://wiki.call-cc.org/eggref/5/s11n). I think there has never been a good solution to versioning then against changes to the code.
Persisting a continuation seems like it's similarly as challenging as persisting an arbitrary closure. Like usual there are things you can't really persist, for example in your call stack you might have some unwind handlers for closing files -- how do you persist file handles?