The posts were published after billionaire Elon Musk bought the platform in October 2022.
But his tweets, which now represent most of the company's communications output, have not mentioned the case.
Antisemitism and Holocaust denial are illegal in Germany.
They also violate Twitter's own terms and conditions.
BBC News has contacted the company for comment.
"Twitter has betrayed our trust," said Avital Grinberg, the president of the European Union of Jewish Students (EUJS), which has brought the civil action, alongside HateAid.
"By allowing hateful content to spread, the company fails to protect users - and Jews in particular."
The case will try to determine whether Twitter is contractually obliged to remove such material.
HateAid legal head Josephine Ballon said: "Twitter assures it won't tolerate violence on its platform. Users have to be able to rely on that."