Not sure about your particular model, but most printers work on a 'how much ink' device. When a cartridge is nearly empty or empty there will be a warning message or symbol, at that point the printer might stop working.
If as you say, you don't use the colour, then just leave the colour cartridge in-situ. But the printer might do a 'cleaning' process, which will use up some ink from both or either one of the cartridges.
Why not run the printer to a empty cartridge, and see what happens. Or alternatively buy compatible cartridges or fill your own?.