Do you ever need DLC to get the Platinum trophy for a game?

No. When designing trophy sets for PS3 games, one of the requirements is that all of the base trophies must be earnable without DLC. Sony will not let developers publish games that do not meet this (and many more) requirements.

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.

Another edge case where your definition of "DLC" may change the answer: episodic games like Walking Dead or Life Is Strange. In these games, you have to play through all episodes of the season to get platinum. If you consider episodes 2-5 to be DLC for episode 1, then this would be another case where you need DLC.