Some PSN titles may run in the future, but not at launch.
If you've been following all the hoopla regarding the PlayStation 4, you'll know that it has a lot of exciting new games lined up. This is an incredibly good thing, since anyone wanting to play his or her existing PSN games on it is out of luck. While Sony has plans to make PSN games playable at a later time, neither the games nor their associated save files will transfer over to the new console.
To begin, PS3 titles - both hard discs and downloaded content - will not be compatible with the PS4 due to significant differences in the hardware's processing capabilities. Eventually, some PS3 titles will make the jump, but only ones that the PS4 can emulate without radically overstretching its processor. As far as PSN titles go, Sony will focus its energies on ensuring that the PS4-specific games and functionality work before providing ways to play older titles in the "longer term."
Given that the PSN library of PS1, PS2, and PS3 games is very extensive and many users have games-in-progress they'd like to transfer, this lack of functionality could be a significant handicap at the system's launch. Bear in mind that these games will eventually become available, but there's no hard timeframe as of yet.