I have a sitemap index pointing to about 8 sitemaps. The sitemaps get updated regularly, but my script only overwrites the sitemap index if the sitemap URLs change, and so the index's Last-modified time could end up being kind of old. Will search engines care? Or should I update the Last-modified header for the index as well?
Your question, as found in the title, is a bit confusing. I am not entirely sure you are trying to modify HTTP headers or just the Last Modified date in your sitemap.
You should update the Last Modified date in the sitemap when ever your other sitemaps are updated. While no-one can say if search engines check this date of just check the header date, I personally would not take chances.
If you are talking about setting HTTP header dates, don't bother. If you update things normally, this will just happen automatically.
As a side note, you do not have to resubmit your sitemap if the file names do not change. Google and Bing will check them just fine. As well, I believe that the request response header will contain a modification date as appropriate and all should be fine. Google, for example, will mostly ignore a resubmit if nothing appears to change. However, I do suggest updating the modify date in your index sitemap whenever you update a sitemap for the simple reason that it can trigger the search engine to fetch the updated sitemap.