The updated answer is "It depends if you consider network passes as DLC".
Many recent PS3 games lock their online content behind a "network pass", a small piece of DLC you need to download from the PlayStation Store but can be redeemed for free by using a one-time code included with new copies of the game. If you bought a used game but want to use the online features, you'll need to buy the code separately (usually for around $10).
If the platinum requirements for these games include online trophies, you'll need that piece of DLC, not all games that have a network pass require online play for platinum, though. Examples of games which require a network pass to platinum are Ratchet & Clank All 4 One and Uncharted 3. Sites like PS3Trophies.org usually list if a game's platinum requires an online pass.
One exception worth mentioning: EyePet has a trophy for downloading content off the store, that content could be free content, but if you don't have access to the store through the game (for example if the store is unavailable in your region or the game's region is different than your account's region), you cannot platinum the game.