Is there any reason to get paper reprints of your articles?

I did it initially so that I would have a nice paper copy of my paper. In principle I could hand these out to my colleagues and other people who were interested in my work.

This is rather out-dated these days. You can just send interested parties a pdf and they can read it on their iPad. Save the trees.

I don't see any reason at all to actually pay for reprints. You have a copy already, right?

In the 20 years since I published my first paper, I have been asked for actual paper reprints exactly three times: Twice by researchers, who each asked for one paper by sending a postcard, and once by my tenure committee, who required paper copies of everything I'd ever published. In the latter case, it was much easier to just download and print new copies than to hunt for the official reprints, which are still hding somewhere in the back of some disused filing cabinet. (My university finally stopped requiring tenure applicants to kill forests about two years after I got tenure.)

So I'm gonna go with NO, there is absolutely no reason whatsoever to get paper reprints.

In Italy, when you apply for a position, you often have to send together with the paperwork one to three dead-tree copies of your $N$ best papers (or sometimes even of everything you have published).

This can easily amount to several hundred pages; multiply it by the number of positions you will apply for. It may be costly and troublesome to print them from a university printer. And, you know, printers are always low on toner the day before the deadline.

So in this case reprints are handy to have. I assume Italy is not the only country where this happens.