Should a graduating doctoral student "care" about his or her advisor's tenure considerations?

I think your approach to this situation is a bit misguided. Yes, you should absolutely care about your advisor’s success (and he should care about yours). But “caring” doesn’t mean that you should sacrifice your own career and self-interest to the cause of helping your advisor, just as, by the same token, I wouldn’t expect your advisor to sacrifice himself for you either. Thus, the way you frame the question as a dichotomy between the extremes of “couldn’t care less” and “will do anything to help my advisor get tenure” is a false dichotomy.

The truth lies somewhere in between those extremes, but broadly speaking I tend to agree with your assessment that the advisor is unreasonable to have an expectation that you will defer the launch of your career for a semester and give up money, health insurance and professional opportunities to stick around and help him finish a project that is no longer a priority for you, for whatever reason. You need to stand up for yourself and explain what your priorities are. But do it in a civil and polite way that shows that you are grateful for the mentoring and other help he has given you (and may yet give in the future through letters of reference etc). Don’t say you don’t care (and better yet, don’t even think it). Do say that you would be willing to make reasonable efforts to ensure a smooth handoff of the project, as you seem to have been doing already. And finally, as @aeismaeil said in his answer, don’t mention the tenure or the grant, since they are simply irrelevant to the discussion. Good luck!

You should have some concern for your advisor's tenure considerations—inasmuch as having a successful advisor helps your reputation as well (helpful if you're a young researcher)! However, that does not give your advisor the right to treat you as a "wage slave" or under substandard conditions claiming "it's for tenure."

I would address this by saying that you'd like to help him complete the paper but you've already made plans to move on that can't be postponed. Don't mention the tenure issue unless your advisor brings it up. It's an extraneous detail—you wouldn't really stick around to work if the paper weren't for tenure, so it's not a germane issue here.

But one other question remains: is there a way you could help out without staying on? Could you help "advise" the replacement, whoever that may be, to get up to speed faster?

Your advisor put himself into this situation. There are a number of very serious mistakes on his part here:

• He's down to one person that knows anything about the paper, and that person didn't do any of the work. This was highly foreseeable; he should have made arrangements to hire someone else and ensure a smooth tradeoff (as you yourself made arrangements to do).
• He doesn't know enough about the paper to take over himself. How it's possible that he doesn't know any of those languages is beyond me.
• He didn't communicate his desire for that student to stick around until they had already made other plans.
• Having realized that he is in this position, he offers you only $5,000. In my (STEM) experience, students in this position are generally offered a post-doc with a salary that, while less than industry, does reflect your professional status (i.e., several times what he is offering you)

There are only two options here:

• If this one paper is so significant that he loses tenure over it, then it's ridiculous that he let it come to this. Frankly, he deserves to lose tenure.
• If this paper is not so significant that he loses tenure over this, then you shouldn't waste time either.

So, my view is that you should care (in the sense of being willing to change your own plans) very little. My answer might be different if you wanted to stay in academia, just because you might need to preserve a good relationship with him at all cost.

If you are willing to take a post-doc equivalent (if such a thing even exists where you are), or if you'd be willing to work on this in your spare time once you start your new job, then you can offer as much. Otherwise, you can just say something like "I understand this puts you in a tight position, but I've been very clear with you about my plans to take a job starting on [date], and I'm afraid it wouldn't be practical for me to stay here for another semester." If he is at all reasonable, he will understand that turning down a good job in your field making good money for a $5K stipend would be crazy.