In a message sent to regional prefects, Interior Minister Gerald Darmanin said an "immediate strengthening" of protection was required.
"The risk of acts being committed is real," he said in the message seen by AFP, especially since "many Israeli nationals are on our territory because of the Olympic Games".
France's Jewish population is the largest in Europe, and the third-largest in the world after Israel and the United States.
France is also home to Europe's biggest Muslim population.
Darmanin ordered police, gendarmes and the military protection force Sentinelle to be mobilised "without delay" to guarantee the protection of "strategic locations frequented by our Jewish fellow citizens, especially during religious services".
Earlier Friday, hundreds of people bid farewell in Qatar to Haniyeh, who died in an assassination that Tehran blamed on Israel.
His killing triggered calls for revenge, and raised questions about the continued viability of any mediated talks aimed at ending the war between Hamas and Israel. Hamas called for a "day of furious rage".
Also on Friday, France urged its nationals visiting Iran to leave immediately.
"Due to the increased risk of a military escalation in the region, visiting French nationals still in Iran are invited to leave as soon as possible," the foreign ministry said.
