The injured were Milon Hossain, 25, Rafi, 7, and Rahamatullah, 11.
Muslim devotees held a rally after juma prayer at Nehalpur Primary School ground.
The protesters later burnt an effigy of Macron, that's when the trio were injured, said Begumpur Union Parishad Chairman Ali Hossain.
Among the injured Milon has been referred to Dhaka Medical College Hospital as half of his body sustained burn injuries, he said.
Last month, French President Macron pledged to fight “Islamist separatism”, which he said was threatening to take control in some Muslim communities around France.
His comments, in addition to his backing of satirical outlets publishing caricatures of the Prophet Muhammad, has led to a social media campaign calling for the boycott of French products from supermarkets in Muslim countries.
Also read: Send back French ambassador, demand Islamist leaders