It is true that when the PS3 boots up - whether you have a PSN account or now - the machine itself communicates with a string of Sony servers, as recently posted on NeoGAF, and a log of applications run on the system is almost certainly transmitted. Similarly, the game you're running at any given point also shows up on your PSN profile, and the chances are that if you run a game from within a "backup manager", then the manager - not the game - will be displayed attached to your PSN account.

According to the terms and conditions of the PlayStation Network, this effectively makes you fair game for a PSN ban of which many variations exist - a timed suspension or even a lifelong ban. The amount of access Sony has to your machine is greater than you probably suspect: the company even has the means to irrevocably disable your console should it so wish, and if that happens, it will remain non-functional whether you're online or offline.