When it comes to poetry, immortality is pretty much right up there with love on the cliché-ometer. A lot of poets seem to be obsessed with the idea of their immortality. So it's really no surprise that the theme of immortality—of life after death—shows up in "Aunt Jennifer's Tigers," too. The speaker imagines that the tigers will continue prancing after Aunt J's death, and thus Aunt J will be able to live on through her art. Sweet. Even though Aunt Jennifer's tapestry might not be as famous as those sonnets by that immortality-seeker Shakespeare, Aunt J still gets to be immortal in her own way. Also, let's see Willy Shakespeare try to sew a tapestry. We didn't think so, Bill.
Aunt Jennifer creates her tapestry out of a deep need to live life through her art and to live on through this art after death. (Also, we're guessing she creates her tapestry out of thread. You got to have thread to make a tapestry, after all.)
Aunt Jennifer doesn't care about immortality at all. Not even a smidge. She's just all about passing the time with needlework, as a lot of old ladies do. The speaker is the one who's all hung up on this immortality business.