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-ctagericity 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 tehories used Mitchel's model where Ded\mu < 2^\mu + absolutness (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.

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.