If I get a "this job you applied for was closed" or even some version of "we didn't pick you," and the sender isn't no-reply@no-job-for-you.com (or whatever) I'll send a "thanks for letting me know" reply because somebody actually took the time to hit send.
Probably.
I'm not a big email person, but these days I feel like it's important to acknowledge honest-to-god attempts at communication.