The artist is just playing with foreground vs background; his body carries on from the open space under the chair, and the 3d nose becomes a curled wire when removed from its normal context.
There's no 'joke' to get, except for your brain getting confused.
The nose still has a surface area after being removed, as you can tell by the index finger disappearing behind it.
Isn't there some message to get? Are you sure? The person seems to be part of some military organization, possibly, going by the costume. So maybe this is some jab at something political that happened with the military around this time? Let me think, when was 'Nam? 🤔 [Looked it up] Yeah 'Nam was going on at this time. (Although the Vietnam war went on for a long-ass time – almost 20 years – so it could have been anything. But still.)
If it's got cultural context, I'm not sure of it, but otherwise it looks like it's just got a theme of incompleteness that the brain fills in similar to some optical illusions like the Kanizsa triangle. The man has no legs, the chair is incomplete, the table's top edge isn't there, the nose.
I do notice some decoration on the collar and shoulders that might point to it being a specific person. Do you know which issue it is from 67?
His take on mid-century Major League Baseball is sort of equal parts adorable and unsettling. Other highlights of the issue: A Houston family is paid $75 to test a fallout shelter for 3 days, canned raw meat, and Chef Boy-R-Dee as mildly exotic party fare.
I also feel like the New Yorker targets issues and trends of its time, so it might be hard to know what this is about with some context of events around the time the issue was published.
I find it interesting that despite everything being all skewed and disconnected, yet there is not one single piece (except the artist name that doesn't count) that is separated from the whole. We are all in this together, whether we want to be or not.
(the only exception seems to be a single arm bulge/shadow thing, but I hope you get what I mean overall)