No, religious conflict has not replaced class conflict as the most important conflict in most modern societies.
There are ways to argue that religious conflicts have become more important than class conflict. For example, a major conflict in the world today is between fundamentalist Islam and the secular West. As another example, there are deep conflicts in the United States today on issues like abortion and gay marriage that are based mainly on religious beliefs and not on issues of class.
However, economic issues (which can be seen as class issues) continue to be the most important issues. The recent American presidential election was decided largely on the basis of which candidate was seen to care more about the issues facing the common person. Outside the United States, the tensions between the Greek government and many of its people are tensions that are based not on religion but on economics. The tensions in China are being stoked largely by inequality among their citizens, not by religion.
In these ways, economics and class are still more important (as Karl Marx would say they always will be) than issues of religion in most modern societies.