The Paris high court issued the order to the popular US micro-blogging site at the request of the French Jewish students' union (UEJF) and others.
Anti-Semitism groups hope to identify the users and have them prosecuted.
Nuno Wahnon Martins, director of European Affairs at Jewish human rights agency B'nai B'rith International, said: "Social networks were created as essentially democratic tools that are also being used by people who oppose democratic principles."
"Like any democracy, the social networks also need to defend themselves, and the first step is to deny those who spread hate speech in anonymity as something to hide behind," he told JTA.
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