This is fake news and just another one of the many press hate-pieces on Facebook, either because they're generally angry that Trump won and need someone to blame or because Facebook has a major part to play in the continued destruction of their industry. There's no indication (even in the original NYT article) that Zuckerberg being annoyed with Cook and him telling his leadership staff to use Android phones are remotely connected.

In fact, the justification for the request (as noted in the NYT article) had nothing to do with Tim Cook's comments, but was rather because "the operating system [has] far more users than Apple". Which, taken separately, is an entirely reasonable request. Why wouldn't you want to have your leadership team experience your product the same way most of your customers do.