Question: I should say that I don't mean splitting a long image onto separate pieces of paper, but rather literally using a very long piece of paper, onto which images/text/whatever could be printed. I am aware this may introduce whitespace at page breaks but I'm more interested in how the hardware would handle it.

I guess the issue is if there is some feed mechanism in most printers which would cause a malfunction with a long roll of paper.

Answer: 
Windows (and Mac, linux) drivers generally work on a "page". The page sizes available in the driver depend on the paper sizes your printer supports. For desktop printers this is A3 (US Tabloid) or smaller. 

Laser printers generally only support those sizes, but some may also support a longer page (banner), typically up to 1.2m (4ft) long. These are all "single pass" printers, where all colours are printed simultaneously. 4-pass laser printers (one colour at a time, hence colour printing is 4 times slower than mono) cannot print banners. There are no desktop lasers that support longer sizes. 

With inkjets it is conceivable that the printer can print continuously, without having to eject the paper. Only the printer manuals can tell you whether your printer is able to. If it is, you still need special software that supports continuous printing.
The majority of the time, the driver or printer itself supports a banner option. If not, your printer may support continuous printing through physical trays or requires pressing a few physical buttons