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.)