I think the reason that this is tough to answer for the PlayStation is that unlike XBL most PS games use their own servers rather than Microsoft's on XBL. Many things that are part and parcel of XBL's platform instead vary from publisher to publisher on the PlayStation.
Because servers are for the most part left to individual publishers to control, there is a rare possibility that player-run servers could fill the gap should official publisher servers go down on the PS3. If that were to happen, and ultimately it is probably unlikely to happen for reasons that I will get into below, it would be possible that trophies would still be unlockable on the PSN account when playing on player-run servers... (Even in the scenario where player-run servers exist it would still be possible for a publisher to request that those trophies not be earn-able or to release a game update that disables those trophies.)
In that case it would be up to individual publishers' sunsetting goals. Historically, publishers have implicitly done what is pretty much explicitly done on XBL: if a product reaches end of life and it's servers need to go down, those portions of it will stop being playable/earn-able. There are many games in history where publishers have fought to keep players from running their own servers after a game has shutdown. There are a few shining stars from when publishers released the ability for players to support a game after its official end of life, but those examples are, unfortunately for us as a community, fairly rare.