Anabaptists (especially of Munster) basically tried to build an officially-classless society. (of course, as is 100% the case when someone tries to do that, some animals got to be more equal than other animals even in Munster, but the sordid details of Van Leiden's excesses and abuses are more in scope for History.SE. You should listen to an awesome Dan Carlin's "Hardcore History" podcast on the topic as a good intro).
In addition, more generally, religions exist on a spectrum where the clergy is more or less separated and defined (including as a class).
In one corner you have religions like in Ancient Egypt, where priesthood were by definition higher caste.
In another, to the best of my knowledge, modern Wicca doesn't have a dedicated clerical class the way Druids existed back-when.
In another, you have Protestantism, which has made the communication between the deity and layperson much more direct than Catholicism or Eastern Orthodox Christianity did, culminating in some branches of Protestantism (especially with Anabaptist branch) where you basically have little need for a priest as a separate class. (I am not an expert, but Latter-Day-Saints seems to also devolve much of clerical stuff on community members).
In general, original Christianity - before becoming state religion after Constanine - has many classless/communist themes, in places. E.g. Acts 2:44. This obviously got perverted to an extent once the Roman Church merged with Roman state, but the concepts of vows of poverty, ascetic monks and such persisted even past that. Cardinal Barbarini may have been upper-class; but a random Dominican monk working the fields or doing some other menial work likely wasn't any different-class from regular peasants, aside from being able to read/write.