Updated with response from Naughty Dog.
Apparently, the game is currently bugged, preventing players from auto-saving the game. Some players are reporting that manual saves are impossible too, once the debilitating glitch afflicts the system. Given that reviewers didn't face this issue when issuing their near-perfect scores for the game, it's likely that the culprit is a day one patch.
Patch 1.01, unavailable for reviewers, ships just in time for the game's retail release today. And that's where all the problems are coming from. IGN has a few potential solutions: "While none of this has been confirmed yet, it appears that staying offline while playing does the trick. For some people, staying offline with the patch installed solves the problem. Others stayed offline and deleted the patch to get auto-save to work."
We recommend disconnecting your PS3 before launching the game. Sure, you'll be playing offline, but the only thing you'll lose is access to the game's multiplayer mode.
Update: Naughty Dog has revealed why the save glitch occurred:
We've addressed the autosave issue, it was related to servers on our end. Please quit to XMB and restart the game. SP and MP are OK to play!— Naughty Dog (@Naughty_Dog) June 14, 2013
Bizarre that ND would run any kind of online authentication on single player save files. Perhaps Will Power's conspiracy theory is true: "Sony added the auto-save bug to TLOU to prove the importance of not being always online."