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Fortunately the central notions of first-order model theory are all absolute (among them $\aleph_0$-stability, stability, superstability, simplicity, NIP, deep, dop, otop, $\aleph_1$-categoricity etc). I don't know any MAJOR applications of forcing to first-order model theory. However there are some: Shelah's original characterization of NIP theories used Mitchel's model where Ded$\mu < 2^\mu$ + absoluteness (later a direct ZFC proof was found). Barwise + Kunnen used forcing to establish some results concerning Hanf-Morley numbers. With AECs the situation is different, Shelah established that categoricity in $\aleph_1$ is not absolute, however I am not certain that even this result should be called MAJOR.