I'm thinking for instance of sci-fi such as Arthur Clarke's Childhood's End and other books of around that time, in which the naive "we'll all get along and have a global government soon" mindset makes the novel feel rather dated. Another instance is GGK's Fionavar Tapestry, and the extremely eighties vibe of his real-world-originated protagonists.

Of course, if you intentionally situate your work in the past, like for instance in Susanna Clarke's Jonathan Strange and Mr. Norrell, or actually even Rowling's HP (albeit the very near past), the feeling aged is less of a problem and can even be an asset as it is for Susanna Clarke.

Another problem with it is that the risks of oversimplifying and misrepresenting the real world events are very big. I'm thinking here of Atwood's Handmaid's Tale - as a dystopian novel, it's fairly unavoidable that it'd involve a fair bit of Real World elements and be presented as the near future of our real world. But anyone who thinks her real-world premise is preposterous, as I did, will have a hard time getting over that to appreciate the essence of the novel.

So I guess I'd say it's a risky business, safer to stick with completely invented things (or an invented world that is close enough to ours to be recognizable, but not so close that misrepresentations or changes will grate, such as in several of Kay's books, and Jacqueline Carey's lovely world). Of course, if you want to make a more direct point about the real world, it's pretty much unavoidable and you'll have to take the risk. And it can certainly be good when done well.

See above, I guess.

Kind of depends, it's fairly straight-forward fantasy in HP and even more so in some other books (Cornelia Funke's Inkspell, for instance). But it can be more crossover or fusion too, sure.