I think Harry already failed his bargaining attempt vs. Lucius, so for flow of story reasons I don't think he's going to call in his life debt, although that makes the most sense of the theories I've seen so far.
I think Harry will bargain with the dementor. "Dementor, if you and your ilk allow this girl to enter Azkaban, I will come and destroy you all. If you refuse to let her enter, I will permit your species to continue to exist." This could be a taboo tradeoff; Harry is trading his sacred value of anti-death-ness to save Hermione.
I think this theory fits in with the story a little better, but it seems less likely to work. It also has a pretty bad failure mode; the dementor ignores Harry because everyone else in the room expects it to, and Harry has to get Fawkes to take him to Azkaban to make good on his threat. (Does Harry know that Phoenix travel is a fast way in? If not, he can probably guess so.)
I was never comfortable with dementors being that impressionable. Harry was able to threaten a dozen of them in Azkaban, and sure, he expected it to succeed; but afterward, that dozen returned back to the main body of hundreds of dementors, all of which then refused to assist dozens of aurors, who would have expected them to. I would argue for at least rudimentary intelligence on the part of the creatures.
EDIT: New discussion thread here.
This is a new thread to discuss Eliezer Yudkowsky's Harry Potter and the Methods of Rationality and anything related to it. With two chapters recently the previous thread has very quickly reached 500 comments. The latest chapter as of 17th March 2012 is Ch. 79.
There is now a site dedicated to the story at hpmor.com, which is now the place to go to find the authors notes and all sorts of other goodies. AdeleneDawner has kept an archive of Author's Notes. (This goes up to the notes for chapter 76, and is now not updating. The authors notes from chapter 77 onwards are on hpmor.com.)
The first 5 discussion threads are on the main page under the harry_potter tag. Threads 6 and on (including this one) are in the discussion section using its separate tag system. Also: one, two, three, four, five, six, seven, eight, nine, ten.
As a reminder, it's often useful to start your comment by indicating which chapter you are commenting on.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: