The first application of forcing to topology was probably Tenenbaum and Solovay's 1971 proof that, consistently with ZFC, there is no Suslin line$^*$. The existence of Suslin lines follows from $V=L$, and so was already known to be consistent with $ZFC$; showing the consistency of the opposite result required "killing off" all Suslin lines with an iterated forcing argument. I believe this was also the first iterated forcing argument.

(The article "Topology and Forcing" by Malykhin (http://iopscience.iop.org/0036-0279/38/1/R02/pdf/0036-0279_38_1_R02.pdf) is somewhat relevant.)

$^*$ A Suslin line is a complete dense linear order with no endpoints, satisfying the countable chain condition, which is not order-isomorphic to $\mathbb{R}$; see http://en.wikipedia.org/wiki/Suslin%27s_problem.