The Yamabe invariant is a real-valued diffeomorphism invariant coming from Riemannian geometry. Using Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a Kähler surface is determined by its Kodaira dimension. We show that the Yamabe invariant of Inoue surfaces and their blowups is zero which demonstrates that the non-Kähler analogue of LeBrun's theorem does not hold.