I have upgraded several desktops and laptops from Windows 8 to Windows 8.1, and have not encountered any issues with printers or anything else. Everything just works, and in some cases 8.1 is more responsive than 8 was.
Try looking up your specific HP printer at the Windows Compatibility Center, to see if there is any specific information on your printer for Windows 8.1. You also might want to check with HP about your specific printer model, to see if their software has some issue with Windows 8 or 8.1.