Unless you can buy a ink cartridge chip reseter I would expect that you will still have the printer telling you the cartridge is empty, I would expect that you could ignore the empty cartridge message and continue to use the printer though.
I have tried compatible inks and refills in my HP printer but I find that none of them are as good has buying the original HP ink cartridges. I found the compatibles to be almost as expensive as the original HP ones, I had print problems and no amount of head cleaning sorted it out. In all the time I have used original HP ink cartridges I have never had to do a head clean!
Also with HP printers the printhead is on the ink cartridge and these were designed in a way that everytime the cartridge was replaced so was the printhead, so you may find that after a few refills the printhead will start to give you problems/will have wore out (my friend had this problem).
My experiences and problems have learned me to just buy the original HP inks now, I know they are more expensive but IMO a cartridge that either doesn't print out correctly or causes you problems is not worth the money saving.