What are nice applications of TatePoitou duality?

Take a look at the recent paper of Mazur and Rubin, "Ranks of twists of elliptic curves and Hilbert's 10th problem" Lemma 3.2, one of the indispensable lemmas of the paper, is a direct application of the PoitouTate exact sequence. 


There are loads of applications and what you consider "nice" hugely depends. One application is e.g. Tate's proof that the Birch and SwinnertonDyer conjecture is invariant under isogenies. That and more applications are contained in Milne's book on "Arithmetic Duality Theorems". 


One of the main themes of current number theory research, instigated by the work of Wiles and TaylorWiles on the ShimuraTaniyama modularity conjecture for elliptic curves, is the study of modularity (or more generally automorphy) of representations of Galois groups. The basic tool for proving modularity is the socalled TaylorWiles method, which involves introducing certain wellchosen auxiliary primes to aid in analyzing a generalized Selmer group (a certain Galois cohomology group) which controls the infinitesimal structure of the space of Galois representations. In applying this method, one relies on a certain formula (due to Greenberg and Wiles) which expresses the ratio of the orders of a Selmer group and the corresponding dual Selmer group in terms of a product of local terms, and the proof of this formula is an application of the PoitouTate exact sequence. See Washington's article in "Modular forms and Fermat's Last Theorem" for an exposition of the GreenbergWiles formula. There are lots of similar applications of PoitouTate in Iwasawa theory. Whenever one studies Selmer groups, one confronts the problem of trying to analyze a global object which is defined in terms of local conditions. The PoitouTate exact sequence provides a relationship between local and global Galois cohomology groups, and so is fairly ubiquitous in the study of Selmer groups. 


I think that Galois cohomology of elliptic curves by Coates and Sujatha is a good place to see some of the applications to Iwasawa theory of mentioned by prof Emerton. 


Three examples of the use of the PoitouTate duality: All parity results (Dokchitsers, MazurRubin, Nekovar, ...) for elliptic curve use this duality somewhere. The duality is also crucial for Euler systems. An Euler system produces a collection of "derived" global cohomology classes. Using the pairing induces by PoitouTate, one can show that these classes are not in the Selmer group but orthogonal to it. This then bounds the size of the Selmer group from above. The PoitouTate duality is an analogue of the Poincaré duality. For instance, Tate reduced the conjecture of Birch and SwinnertonDyer over a global field of characteristic $p$ to the finiteness of Sha. This argument uses Poincaré duality on the corresponding ellitic surface over the finite field. Analogous in Iwasawa theory, one uses the (Cassels)PoitouTate duality in the computation of the leading term of the characteristic series of the Selmer group of the elliptic curve. I would suspect there are more examples in "Cohomology of Number Fields". 

