Try an exact phrase web search for "NASA shows the film as part of its management training program. Prospective managers are asked to find as many inaccuracies in the movie as they can." You'll get lots of hits. Lots and lots of them. That smacks of hand-me-down news, which makes the story ring a bit false. Where a source is attributed it's the Feedback section of the September 1, 2007 edition of New Scientist:
WHY does NASA show the movie Armageddon as part of its management training programmes? ... In reality, the screenings are just a game for NASA's space geeks: who can find the highest number of impossible things in the movie? The record, Feedback is told, stands at 168.
The ellipsis is a few fluff sentences I edited out to keep my quote within the realm of fair use. I didn't cut that much. Note that there's no attribution, no context. It's a fluff piece. Who knows whether it's true?
That said, I can think of one context where this might well have happened. NASA does send their management trainees to fairly extensive and sometimes intensive classes. Occasionally they need to do something fun to lower the intensity. Some companies hold a paintball session to accomplish that end. The "fun" session at a NASA retreat is reportedly a bit more nerdish. Tearing a "so bad it's good" sci-fi movie to shreds would fit right in.
The New Scientist article was printed 6½ years ago. If the article was true, there's no knowing if it was something NASA still did in 2007. It might have been something NASA did years before that, but the news came to New Scientist by word of mouth. I highly doubt it that this is used now because (a) the movie is now old and (b) management everywhere regularly changes how they do training. Now they might rip Gravity to shreds.