I was wondering about that. I was thinking that as long as you don't harvest an ancient plant, that one last time, it should remain forever. I don't know, and I am afraid to just leave an ancient plant un-harvested to see if it is immortal.
I haven't seen an Elder plant with any kind of need or pests. So I suppose that in theory, and being very careful with the X button around it, you could keep it alive forever...