Short story: fail.
HP has good support on Linux, and Open Source drivers, through the hplip project. This printer is also supported through the hplip project. However, it requires some propietary firmware (plugin) to work.
The Raspberry Pi is an ARM device. Most consumer PCs are x86 or a deviation thereof. So the drivers and the firmware is made for an x86 architecture. This is not a problem with the driver, as it is Open Source. But the firmware is closed, and not available for ARM.
The end of the story is that the HP Color LaserJet 1600 does not work on the Raspberry Pi, though some other HP printers does.
Open Source for the win! Close Source FAIL!