Well I poked around at this a little bit. CF does not appear to have any RSS
feed functionality so that's out.
You could subscribe to threads and in your "Control Panel" set it up to get instant, daily or weekly digests. That could be a crap-load of emails...
The third way is go to Control Panel
and set for max messages per screen
(40) and linear view.
Then open a thread and under thread tools hit "printable version" - this pops open a new page with 40 posts (current page) - save it as text and then hit the next page.
Bit of a PITA but basically doable that way. Of course you have no option to reply to the messages offline and say upload later.
Not sure how any of this would translate to the mobile app.