There is a considerable literature on `applications' of Dowker's result to sociology. This is sometimes doubtful in its depth! The development was started by R. Atkin.
As an example look at
As an offshoot of this there is fairly recent work in discrete maths (see work by Hélène Barcelo). I will not try to describe this other than saying it looks at an idea of the connectivity of a relation.
Back in the world of algebraic topology, it provides a way of proving that the pro-object in the homotopy category of simplicial sets, that is given by the Cech complex construction is in fact homotopy coherent. This provides a way of linking strong shape theory to the original form of shape theory. (It is not hard to prove this coherence directly although I only know one proof that has been written down in a thesis of one of my ex-students.)
(Edit: I forgot another example. You start with a group, $G$, and a family of subgroups, and ask to what extent invariants of the family give you invariants of the big group. This was the subject of a paper by Abels and Holz (Higher generation by subgroups , J. Alg, 160, (1993), 311– 341.) The family generates a covering of $G$ by its cosets. The two complexes given by that covering allow proof to be shortened and also in certain circumstances for links to Volodin's alegrbaic K-theory to be given.)