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
http://www.ehu.es/ccwintco/uploads/1/11/Blanca-Cases-Q-analysis.pdf
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.)