This morning, I disconnected the PS3 from the network, ensuring it was offline, and did some tests.
I tested numerous games I'm 99% sure we're PS+ offerings and every single one loaded without issue, including one of the Metro games I hadn't ran yet. It simply installed the trophy data and got to the main screen. Other games tested include Dragon's Dogma, ICO, and LittleBigPlanet Karting, among others. No authentication ever appeared to be happening, so my best guess at disappointment is if my PS+ expiration date is saved somewhere on the hardware as a ticking time bomb once it expires. No way to test that till it happens, but I don't believe that form of authentication makes sense for an online support structure.
The most annoying issue I ran into was while playing a purchased PSN game, Bionic Commando Rearmed. After each level, it wanted to connect to the network to upload stats. Easy enough to press O and back out of it though.
Sadly for me, the version of Wet I have is merely the demo. It doesn't tell you that till you hit the main screen. Given the cost of it now, that's unfortunate. I wouldn't expect demos to be impacted by authentication concerns.
Best advice -- Download everything from your old list and disconnect your PS3 from the network well in advance of the store closure next July.