Religious belief played a role in both of these revolutions, mainly in the form of a desire to be free from governments which interefered with and dictated people's religious behaviors.
In France, part of the revolutionary movement was anticlerical. It was a rebellion that was, in part, against the Catholic Church. This was not so much a doctrinal or theological rebellion. Instead, it was a rebellion against the privileges the Church enjoyed and the role it played in reinforcing the power of the oppressive monarchy.
In America, colonists were also worried about the imposition of religion. There was alarm in the colonies, for example, when colonists came to believe that Anglican bishops were to be sent to America. They worried that this would lead to a situation in which Anglicanism was imposed on the colonists. Many of the colonists were of other religions and did not want this to happen.
Colonists also worried about the Quebec Act. In this law, the British government seemed to be privileging Catholicism in the newly-taken area of Quebec. The colonists saw in this another attempt to impose hierarchical religions like Catholicism and Anglicanism on the colonists who had, particularly since the Great Awakening, been likely to hold more egalitarian religious beliefs.