I don't think it makes a big difference—the printer caps the printhead when not in use to prevent it from drying out regardless of whether it is on or off. What matters is that you avoid cutting power to the printer, especially while it is busy—this could leave the printhead in an uncapped state which will more often than not lead to printhead clogging and (in severe cases) possibly failure. (Trust me, I learned this the hard way with Epson printers . Unlike HP printers, Epson printers tend to treat carriage jams as fatal errors, typically with error codes 0xEA or 0xE8, which require a power cycle to recover from. This results in the printhead becoming uncapped for an extended period of time, causing severe clogging.)
For printers that are not being used regularly, I'd generally recommend that the printer be power-cycled every week or so (though this can be longer for HP printers, up to several weeks at a time) as doing this typically triggers a maintenance cycle that helps keep the nozzles clear. This process does consume some ink, and because of this need for maintenance (which HP calls servicing), inkjet printers are generally best for regular printing. (HP printers are better than most over long periods of non-printing; see this Wirth Consulting study for details.)
Printers generally do not turn themselves on for servicing. The HP PageWide printers do service themselves when powered on and idle; see the PageWide Technology whitepaper for more information.
This HP article on how ink is used contains relevant information on ink usage in printhead servicing.