It can be argued that a major fallacy that Zinn includes in his presentation of history is through his use of materialism to dictate the unfolding of history. From the opening chapter, Zinn makes it clear that his read of history is predicated upon a dialectical materialist notion where there are those in the position of economic power that sought to exploit those who lacked economic power and control. The fallacy here is that not all of human interactions necessarily can be reduced to economic notions of the good. Zinn presents Columbus and European settlement of the "New World" as an entity whereby materialism was the sole driving force within it. Obviously, this can be disputed. For example, there were more psychological motives that played a role in the founding of the New World.
Essentially, the fallacy that Zinn features is that all human interactions can be reduced to materialistic control. Zinn make the argument that power reduces to economic control. Those who had it exerted it against those who lacked it. Zinn's retelling of history, his idea of the "people's voices," comes from this idea. While it is presented in a compelling manner, a fallacy is to presume that all interaction systems are rooted in money and economic progress. In this, one can make an argument as to a potential fallacy. Its logic is sound if one accepts the presence of materialist interaction underscoring all human interactions. I think that this acceptance can represent a logical fallacy.