I’m saving the ultimate, unimpeachable reason for last, but let’s get right to it:

1. It’s Not a Christmas Movie: Anyone claiming it is a Christmas movie is either having us on, or they’re as clueless as Deputy Chief Dwayne T. Robinson. That’s enough to settle the matter, but I’ll go on.
2. F Words and Nudity: If a certain word starting with “F” is in the movie, it’s not a Christmas movie. Also, if you are a free, American adult, you have a choice of watching Christmas movies or watching movies with topless girls. What you cannot do, by the immutable laws which govern the present universe, is have both in the same film. You know why. “But,” you might respond, “I saw an edited version once! So that version was definitely a Christmas movie!” Well, first of all, see #1, and secondly, how can it be a Christmas movie, if it can’t be shown on an airplane without edits? I’ll calculate the last digit of pi as you struggle to craft an answer.
3. Year-round Viewing: Suppose, on a lazy evening in August, you call a friend of yours, and inquire what they’re up to. “Watching Die Hard,” they reply. What is your response? You’re not going to say “In August?” or “But it’s not Christmas!” No, you’re not going to say that. And you know you’re not. QED, not a Christmas movie. Die Hard, being an action flick rather than a Christmas flick, is something people are likely to watch any time they want some action-fueled escapism. If you come home, and your roommate is watching Die Hard, you don’t have to check the calendar. If your friend were watching It’s a Wonderful Life in April, you probably would remark on their viewing it “out of season,” even though it’s a fine film for all times of year, and only the finale is set during Christmas. But it’s so much closer to being a Christmas movie than Die Hard, that most people would remark on the perceived oddity of viewing IWL in summertime. Also, see #1.
4. An Action Hero Does Not a Baby Jesus Make: I needn’t get so deep (because see #1), but I suppose someone will propose that Die Hard is a film in which all seems lost, until a hero arrives, to set all things right. Messiah McClane enters our darkness, punishes the wicked, and redeem the captives. Why, Nakatomi Plaza might as well be a stable and a lowly manger. Well, OK, but now you’ve made every action film a Christmas film.
5. And now, the final point, and the one that will force the Die Hard diehards to their knees, and force them to confess the truth. Were this a Christmas movie, you know those Japanese guys would be eating some KFC. Are those Japanese guys eating KFC? No, they are not.

If that last point doesn’t clinch it for you, I don’t know what to tell you.