Problems with Mac OS 8 and HP printers were noted from the very release of Mac OS 8 (see Troubleshooting Mac OS 8 item). In general, the solution to these problems has been to wait for the next update to the HP drivers. Based on reader reports that I continue to receive, the latest drivers have still not resolved all existing problems.
Most commonly, reports indicate that having background printing enabled prevents printing. For example, Guy Samson could not get background printing to work on his LaserJet 5MP (using driver version 8.3.1) connected to a Power Mac 7200. Roy Chen specifically identified HP Backgrounder (from the 9.2.1 HP DeskWriter 600 Series drivers) as the source of his problem in this regard (using a PowerBook 3400). Disabling the extension (and perhaps HP Print Monitor as well) allowed printing to proceed.
Harris Mirkin claims that the 6.0.3 version of the DeskWriter 500 Series drivers would not allow printing of any sort with his 68040 Centris 610.
By the way, a quick way to check for the latest versions of all HP drivers is to use Version Tracker and search on the term "HP." In this case, it shows that there is a 9.2.2 version of the DeskWriter 600 driver. I am not sure if this fixes Roy's problem. Otherwise, the drivers listed above are the latest ones.
Of perhaps related interest, a report on MacCentral awhile back claimed that Apple had confirmed a background printing problem with the StyleWriter 1500 and Mac OS 8.