You could end up paying a lot more for ink in the long run by trying
that. Aside from the thermal qualities, the shade of ink will
probably not be exactly the same. A different shade of ink will
contaminate the felt pad (whatever material it is made of) on the
printhead where the ink cartride mates with the printhead. Once that
is contaminated with the wrong shade, you will have to run a ton of
ink through the printhead to get rid of the contamination. Sell the
left over HP ink on ebay or dump it and buy ink specifically for the
i560.
You could end up paying a lot more for ink in the long run by trying
that. Aside from the thermal qualities, the shade of ink will
probably not be exactly the same.