The Officejet 8710 is among the HP printers that cannot print if any ink cartridge is empty (there is no override provision as some printers have). If any color cartridges are low, you can set it on black-only to minimize the use of color, but the color cartridges cannot register as empty.
Per the HP Customer Support Knowledge base (excerpts):
The printers listed in this document cannot print when one ink cartridge is empty.
Your printer is designed to print only when all ink cartridges have ink. Some ink from all the ink cartridges is used in periodic servicing tasks. This helps to prevent issues such as printhead clogs and poor print quality. There is no way to bypass this feature.
When an ink cartridge is depleted, you must replace it before you can print.
If one or more color ink cartridges are low but not empty and you want to minimize the use of color ink, change the settings in the HP software to print with black ink only (grayscale). If you print with black ink only, the other ink cartridges must have ink for the printer to print.
There is more explanation at that link.
HP's position on the printer not operating with an empty cartridge is based on a premise that the printer could be damaged otherwise. So there is no way around that without hacking the printer (which would void any warranty, would be very difficult to do since the firmware is proprietary, and stands a good chance of totally ruining the printer). The printers are inexpensive, and nobody stands to profit from investing in figuring out how to do it. So I wouldn't expect to see any kind of workaround for this.
However, there is potentially a way to get it printing again at less cost than buying HP ink cartridges. The OJ 8710 is among the HP printers that employ a security chip. This locks out the printer if it detects a "counterfeit" ink cartridge (which is for your benefit to ensure that you get the very best printing experience from your HP printer). :-)
For some printer models that didn't originally come with this "feature", HP implemented it surreptitiously through a software upgrade. There was heavy pushback, threats of legal action, etc., and HP decided to rethink its policy. They now have an optional software upgrade that can be manually downloaded from HP if you know about it, which will turn off this verification and allow the use of non-HP or refilled HP cartridges.
Just a warning, some other printers have an option to turn off the check for OEM ink cartridges. When you do it, it disables a lot of advanced features in the printer driver software, and they claim that if the printer malfunctions, you're on your own. So they let you do it but strongly discourage it. I don't know if that applies to your model, but you might want to inquire.
This info page from HP describes the use of the "security feature" and states:
HP believes in fair competition and does not employ technology preventing the use of refilled and remanufactured cartridges. Our printers with the dynamic security feature may not operate with a cartridge that does not include an Original HP chip.
This seems to be saying that the feature only addresses third party cartridges manufactured from scratch, with a non-HP chip. That would still leave the issue of original or refilled cartridges that claim to be empty based on character count, and prevent printing even though there's still ink left in the cartridge. But at least you could buy a new, unneeded fake cartridge for less than the cost of a new, unneeded HP cartridge. Ah, the joys of inkjet printers.
But here's a link to the HP Customer Support Knowledge Base that describes turning this off for your printer. However, your model is included in a group where manufacturing date appears to determine whether you can disable it:
this update disables dynamic security for printers manufactured before December 1, 2016. For printers manufactured on December 1, 2016 or later, dynamic security remains functional with this update.
My reading of the information is that for printers manufactured earlier, the provision wasn't clearly stated, so HP is letting people disable it. However, ones manufactured later clearly stated the policy on the box or in the user manual, so buyers were forewarned and out of luck.
I found online discussions saying that HP re-reconsidered, and other articles saying no, not really. So it looks like if the printer was manufactured before December 1 2016, you can probably disable the security chip feature and use less expensive third party cartridges. If it was manufactured later, maybe it can and maybe it can't. The situation is clear as mud. But that link will at least get you to the right place to find out; maybe just try it.