I have no problem with my words being on the Hacker News and available for everyone to see here. Obviously I wouldn't post them otherwise.
I do have a problem with people taking those words and redistributing them elsewhere. Just because it's on the Internet doesn't mean it's free, and I fail to see how what we're talking about here is much different to this case that was doing the rounds on the tech forums the other day:
Just because there is a reasonable reuse of the content doesn't mean that all reuses of the content are reasonable. Also search engines and archive.org preserve the context and content.
Caching is an interesting case, or rather, several interesting cases.
If it's not a faithful representation, i.e., if it does not reproduce anything exactly, omits any part of the material, adds additional material, or is out of date, then I'm not in favour. It's not really a cache at all at that point, and such flaws are obviously potentially damaging to both the visitor and the original host service/content providers. The only likely reasons I can see for not reproducing faithfully are incompetence or active leeching. (I note in passing that I have never seen a cache or archive web site that actually did reliably reproduce content faithfully. All of the major services, including heavyweights like Google Cache and archive.org, failed badly on this criterion last time I checked them out. But real cache proxies tend to pass, as long as they refresh at an appropriate frequency.)
If it's faithful but continues to make content available after the original host has pulled it, then it's potentially valuable to visitors and probably many hosts/content providers would have no objection, but I think it should be opt-in. Not respecting a service that puts content up for a while but then chooses to take it down again for whatever reason could have a chilling effect on willingness to put the content up in the first place, and could undermine business models that would otherwise be useful, reasonably fair to all parties, and financially viable. (This point doesn't really apply to sites like HN, though, as they typically publish posts indefinitely anyway.)
If it's faithful and timely but still robs the original host of valuable meta-information (notably real server logs that have at least two legitimate uses: helping to optimise the site based on real user behaviour, and supporting claims made to third parties about site traffic) then this could also be harmful to the host service, but on the other hand, it's not clear to what extent such meta-information should ever be relied upon anyway given the architecture of the Web today. Ideally, I think we would have some sort of standard proxy notification so that caches and such could forward relevant meta-information to the original host in some sort of digest form. That way, this one becomes a non-issue, as long as any cache/archive service implements the appropriate notification to be fair to the original host.
Basically, I think cache/archive services can be widely useful and probably many hosts/content providers would have no objection, but given that they can have significant downsides, they should always be opt-in and ideally we would have simple conventions based on something like robots.txt to indicate this.