If Dementors were just dangerous evil creatures roaming out there and occasionally killing unlucky people who didn't Apparate away in time - like trolls, or dragons - Harry wouldn't care much about them and their association with Death.
"Humanism" suggests he'd technically want to eliminate them anyway, and as it turns out this desire seems wise. Dementors can breed (asexually) if whatever makes decisions for them decides to prepare for more victims. It happened in canon. In the absence of any Patronus 2.0 this seems like an existential risk or a risk of an unpleasant future for humanity, depending on how the Dementor decision maker works. And again, given the existence of Dolores Umbridge with her canon use of Dementors, I see no reason to believe that every present and future Ministry official will refrain from offering them Muggles. Nor does it seem wise to wait until Dementors start multiplying before reducing their numbers.
Now, Harry doesn't know all this, but "hole in the world" sure sounds like a possible threat to humanity's existence. He must believe in a non-zero risk from Azkaban. Logically Harry should ask if this risk exceeds the Star Goat or Anti-Pascal's probability. (Example: the chance of cryonics working, in any particular case, seems to far exceed the Star Goat probability of it interfering with or preventing an afterlife.) And neither his pessimistic mentor nor his conditionally pro-death mentor seems to consider the chance of ending humanity through killing Dementors worth mentioning.
Update: Discussion has moved on to a new thread.
After 61 chapters of Harry Potter and the Methods of Rationality and 5 discussion threads with over 500 comments each, HPMOR discussion has graduated from the main page and moved into the Less Wrong discussion section (which seems like a more appropriate location). You can post all of your insights, speculation, and, well, discussion about Eliezer Yudkowsky's Harry Potter fanfic here.
Previous threads are available under the harry_potter tag on the main page (or: one, two, three, four, five); this and future threads will be found under the discussion section tag (since there is a separate tag system for the discussion section). See also the author page for (almost) all things HPMOR, and AdeleneDawner's Author's Notes archive for one thing that the author page is missing.
As a reminder, it's useful to indicate at the start of your comment which chapter you are commenting on. Time passes but your comment stays the same.
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: