PARIS—A French criminal court Thursday ordered a probe into whether International Monetary Fund chief Christine Lagarde was complicit in any misuse of public funds in 2008, when she was France's finance minister.

The Cour de Justice de la République, a special court handling allegations criminal matters involving ministers, was responding to a claim by Socialist Party lawmakers that a €420 million ($602 million) award to tycoon Bernard Tapie that was done under her watch was unjustified. The award followed a complex case...