I can confirm that the door close button works at least some of the time. I used to ride an elevator every day on the way in to work and was surprised to hear about that button being a sham, since my general impression was that it worked. So, I did a test.

I started timing how long the door would take to close on all of the solo rides I took on my regular elevator, alternating hitting or not hitting the button. The results confirmed my intuition that the button did speed things up significantly. I've forgotten the exact numbers but the increase was pretty large, something like 5 seconds vs 15 seconds, if I hit or didn't hit the button.