Three examples of the use of the Poitou-Tate duality:

All parity results (Dokchitsers, Mazur-Rubin, 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 Poitou-Tate, 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 Poitou-Tate duality is an analogue of the Poincaré duality. For instance, Tate reduced the conjecture of Birch and Swinnerton-Dyer 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-)Poitou-Tate 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".