My naive opinion is that this question factors into two questions:
How do facts from algebraic topology shed light on posets?
How do facts about posets shed light on number theory?
The second question is, I think, a little easier to answer. The poset most obviously relevant to number theory is $\mathbb{N}$ under divisibility. In some vague sense the study of this poset is equivalent to the study of Dirichlet series (see for example this MO question). This perspective is, I think, due to Rota.
The first question is not about number theory. Once you realize that you can associate an abstract simplicial complex to a poset in a canonical way then I don't think this idea sounds so strange; it is commonly used by combinatorialists (see for example these notes). There are also some motivational theorems in this area, e.g. every finite CW-complex is weak homotopy equivalent to a finite topological space, and the category of finite $T_0$ spaces is isomorphic to the category of finite posets, but I'm sure an expert could say more here.