The reason for the use of "veneer" is covered by other answers (a superficial coating applied to something for appearances, used here metaphorically), but I'd like to add a possible perspective on the headline writer's choice of "puncture".
There is a legal phrase "piercing the corporate veil", which is in widespread enough use in corporate/financial/legal areas that I do not doubt the headline writer is aware of it. (This is the New York Times, after all.) "Piercing the corporate veil" refers to a process by which the distinction between the corporation and the people running or owning it is blurred.
Under normal operation there is an official legal separation between the corporation, as an independent legal entity, and the private individuals who own and run it. But part of maintaining this "corporate veil" is that CEOs and company owners have to maintain that distinction between corporate property and actions and personal property or actions. If they don't - if they act through the company for their own personal ends (and explicitly counter to the benefit of the company) - courts can "pierce the corporate veil" and hold them personally responsible for corporate actions and debts.
I have no proof of this, but one possibility is that the headline writer for the New York Times was attempting to bring in this phrase as a metaphor by the choice of the word "puncture". (It's certainly where my mind went to first.) Like the corporate veil, the office of the President has something of an unofficial separation between the position of the President and the person who happens to be the President. For example, with Presidential civility, while the person holding the office might have rather scathing personal opinions about past Presidents, the precedent is not to use the office of the President as a platform to broadcast those personal opinion.
That is what I'm thinking may be behind the choice of "puncture" - the headline writer may be attempting to imply that Trump's actions blur the distinction between his personal opinions and positions/behaviors held by the office of the President, by using a metaphor that only makes sense if you're up-to-speed on financial legal terminology. (As I said, this is the New York Times.)
Mixed metaphors are very headlinese, so "puncture veneer" is a short way to smash together a bunch of relevant concepts.
P.S. Why "puncture" rather than "pierce"? I'd say "pierce" has a more deliberate connotation to it. "Piercing the corporate veil" is an action taken by the courts to remove the distinction between corporation and private person. On the other hand "puncture" conveys more of an accidental nature. Trumps actions weren't intended to remove the distinction between the office of President and the person holding it, but may have done so anyway.