What are the major applications of (set theoretic) forcing in model theory?
Rather than addressing the issue of whether an application is major or not, let me point out a recent one: See
(The paper is downloadable from either of their webpages). What they do is work in the context of Abstract Elementary Classes, and provide a unified framework for several results on the number of models of size $\aleph_1$ in various infinitary logics. Their approach uses "$\mathbb P_{\mathrm{max}}$techniques" to obtain the models through iterations of countable transitive models of enough set theory via embeddings given by the nonstationary ideal on $\omega_1$. The models so produced are typically illfounded, as the technique is applied without the assumption of large cardinals. Similar constructions appear in the followup paper
(Also available at their websites.) Actually, recent work on Abstract Elementary Classes provides a nice environment for nontrivial interactions between model theory and set theory, not just in the form of forcing techniques. See for example this post at the Forking, Forcing and back&Forthing blog. 


Applications of forcing in model theory, following directly from Robinson's 1970 work, are reviewed by T. Zhang, Modeltheoretic Forcing and Its Applications. For a more recent application, the construction of finitely generated models, see A.O. Houcine, Forcing in Model Theory revisited. 

