I think it is obvious we should care, in the sense of being benignly interested; but should we care in the sense of changing our actions in order to design changes in the world post 2100? I don't think I've written on this directly (as I say in reply to DB). Morality and economics discusses mt's The Seventieth Generation, which would be relevant, but sadly I'm looking at a different perspective. I say don't think more than 100 years ahead in reply to NB in 2014 in Meesc, and the discussion continues for a while there.
There are two valid reasons why you might not care about post-2100 (or post-2117; or some other arbitrary but distant time. But not a time as close as 2050). One is that you like to use a conventional-economic discount rate of X% and that reduces future concerns so that mathematically and economically you're convinced those concerns are so deflated by discounting as to be uninteresting. And two is that you think our ignorance of those far-off times is so deep that we cannot possibly usefully design our behaviour to helpfully shape their world. Those two reasons aren't totally separate, but for various reasons I'll talk explicitly only about the second.
Not to ruin the tension, but my answer is a qualified no; we should not care-as-change.
And why would I think a thing so manifestly absurd? Because our ability to foresee the future is so weak. An easy recent example that comes to hand is the IEA (and everyone else's) inability to predict solar PV growth even a year ahead, let alone a decade or a century. You could plausibly say that case is hard, and that broader trends are easier to forseee, but meh. Would we have thanked people in the past for trying to see 100 years ahead? When I asked that before, Gavin replied "Central Park" but I wasn't convinced; and I'd add that on the scale of GW, that's trivia. Dunc did better with "the London sewer system" but again; it's a small thing.
And secondly, too much striving to foresee and manage the future leads to too much managing, which is bad, in my opinion. This of course leads back to Hayek, but I see I've failed in my duty to provide the promised posts on him; Hayek and Climate provides a reasonable sample. I don't mean the comparatively minor parasitic class that flies off to the various political climate conferences around the world. I mean more the encouragement of the very concept that it would be a good idea if the state did more planning, when it should be doing less.
"Not caring" doesn't include not doing sensible and obvious things. One of which is to Do Science, which apart from anything else is cheap. The science we've done so far leads us to conclude that Sea Level Rise will be about +1m by 2100. You can make a case for only 50 cm and you can make a case for 2m. But - barring some major revision - it won't be as much as 10m (which would be disastrous) and it won't be as little 10cm (which would be too little to notice). So that's a happy co-incidence in a way; there's no obvious reason why SLR out to "about as far as we can usefully look" should sit just around the "irritating but manageable" level. You'll notice I've spoken only of SLR, because it kinda fits my narrative. But I could spin similar words around temperature change I suspect. Ecological response as usual I leave to others. To get with certainty to clearly disastrous levels of SLR you have to go out ~500 years; and that's too long. Can you help the case by replacing certainty by "at least Y% probability of"? Doubtful.
* So What Climate Change Stories Would Sir Philip Sidney Tell asks Eli. Somewhat disappointingly, it isn't about predictions from that era. I prefer the Python version.
* Freeman Essays #4: “Capitalism: Who Are Its Friends and Who Are Its Foes?” - old stuff from CH
* Dangers Lurk in Timid Defenses of Free Trade - sort of more old stuff, from CH