Also note, that despite advice given below by rest of the thread, legal does not care whether you store it yourself, or pull it via API; Copyright / republishing permissions are required for the content either way.
(which brings you another issue, even if you gets lots of people agreeing with you that it is ethical (which I guess you will not get), is it ethical to do it expressly against the wishes of individuals?)
Cause a lot of websites do that and they never ask any permissions. That would turn the Internet into a permission hellhole. Unless your data is not accessible it would be replicated.
I don't believe your issue would be with HN. HN is just an aggregator for content published elsewhere, so they have no say in whether or not you also link to that content.
As regards wanting to publish 'interesting threads', by which I assume you mean debates in the comments; again, I wouldn't worry too much re HN themselves. There are no T&C's to be agreed to when you register an account here, so no-one who comments has granted HN exclusive rights to publish, or transferred copyright on anything they write here. Nor has HN asked for, or claimed any such rights.
So, if I hereby also grant you non-exclusive rights to publish my ill-informed drivel elsewhere, then that's a matter between you and me. Nowt to do with HN.
Of course, in theory, this would mean that you'd have to satisfy yourself that anyone else who comments here and whose words of wisdom you want to reproduce is similarly amenable –and there you open a whole different can of worms, as regards whether or not a comment published on the intarwebs, with no copyright claimed or given, is in the public domain, or not.
I would suggest it [at least partially] is, as is borne witness to by the common practice of reproducing other people's tweets, as practised by most news websites.
What I COULD see as a useful, ethical, and legal service would be something akin to LWN, i.e. you publish LINKS to threads and to particularly salient comments, and add an editorial writeup of the discussion.
This seems like the sort of thing people would be able to sue you for. As far as I can tell, Hacker News doesn't do anything special with copyright of comments. As with all text you write, as soon as it becomes "fixed" by being recorded, the author has the copyright to it. Since the author decided to publish it in a Hacker News comment thread, that basically implies that Hacker News has permission to copy their work and publish it in the standard ways that it does.
Now there is the concept of "Fair Use". Hosting interesting comment threads may or may not be "Fair Use", but either way that's the sort of thing you find out by getting sued and seeing if it works as a defense. If you host it without any commentary, that's less likely to qualify as Fair Use than if you summarize stuff, say what you like about the thread, add your own commentary, etc.
Hopefully an HN moderator or similar will know the answer and reply here. If they don't, you should email HN.
In that case, you're just curating and displaying HN content without hosting it.