I was very bad today and printed after being warned by the software that my printer will be damaged. It is a Canon dot matrix printer with four ink cartridges.
A dot matrix printer? Seriously?
Assuming you meant inkjet, if it doesn’t have enough ink the element that heats the ink can overheat and your printer will go :mad: then 
and won’t print anymore.
While technically possible, it almost never happens. and, if it does, it’s because it is a cheap printer for which it would be cheaper to buy a replacement printer (with ink included) than to just buy new ink.
Why yes, I typed in inkjet and the keyboard sent dot matrix.
What model of teletype are you using?
Punched tape with a optical reader, steam powered, why do you ask?
Your inkjet may be a dot matrix printer. It’s just not a conventional impact dot matrix printer. The number of people not viewing a dot matrix image of this post right now would be very small, possibly zero.
What, nobody else reads the Dope on a Tektronix vector display?
OK, neither do I
The other issue is that the ink feed can start to suck air, and thus get a bubble in the system, which pretty much wrecks it. Not so much damaged, but it becomes unusable until you can work out how to reprime it. With gluggy inks this may not be easy or even possible.
Sending it through that time warp without enough ink would damage just about any printer.
I would suspect that it damages the printer in the following way.
The fundamental purpose of a printer is to sell ink. The printer will not be working towards this purpose with optimal efficiency if you wait until the ink cartridge is empty before you buy more ink.
The Canon printers use the same printer head forever and only the cartridge is replaced. Running out of ink means the remnants dry in the printer jets. HP replaces the printer head with the cartridge so it wouldn’t matter for that brand.
Truth.
Low on ink, well, it is still printing fine.
Out of ink, still printing.
You must press the reset button to over-ride the warning, still printing.
Mr software writer, have you ever heard of the story where a boy cried wolf, no,I didn’t think so.