I was reading this:

http://manuals.playstation.net/document/en/ps3/current/game/hddinstall.html

...and I'm not sure that I quite understand it. Can I install a PS2 game on my PS3, and then play it without the disk? That's what it sounds like from the user guide, but I don't see how Sony could stop piracy.

FFXI installs completely on the hard drive and you no longer need to use the disk.

Thats only for the hand full (FF) of PS2 games that used the HD that was available with the First Gen PS2. No you cannot take a PS2 disc and install it and run it from the HD on the PS3
Edit..other than the FF game

no....

There was a harddrive for the ps2.
A few ps2 games installed to the hardrive. The instructions your reading allows you to play those games on your ps3.