$\mathbb CP^2\#\overline{K3}$ (trivial SW because it is a connected sum with $b^2_+>0$ for both pieces). This is an example without having to use the Bauer-Furuta invariants (contrast with Kyle's comment), but with the SW invariants of the 4-manifold's *opposite* (reverse $X$'s orientation into $\overline X$). I learned this example from Tim Perutz at some point: The symplectic manifold $X=K3\#\overline{\mathbb CP^2}$ has $b^2_+=3$ and $b^2_-=20$ and $b^1=0$ and nontrivial SW invariants (note that $b^2_++b^1+1$ is even, this quantity is the parity of the various SW moduli spaces and needs to be even for the ordinary SW invariants to possibly be nontrivial). By doing Fintushel-Stern knot surgery on K3 we get exotic copies $X_K$ (knot $K$) distinguished by the SW invariants. Then $SW(\overline X_K)=0$ because $b^2_+(\overline X_K)+b^1(\overline X_K)+1=21$ is odd, with each $\overline X_K$ homeomorphic to the non-symplectic $\overline{K3}\#\mathbb CP^2$. So the "technique" of orientation-reversal should be added to our arsenal -- related discussion is found in Draghici's paper *"Seiberg-Witten invariants when reversing orientation"* and its references, for example.

More interestingly (perhaps), Taubes studied a possibly infinite set (if not all diffeomorphic) of smooth 4-manifolds $\lbrace X_K\rbrace_K$ all homeomorphic to the non-symplectic $3\mathbb CP^2\#23\overline{\mathbb CP^2}$, obtained from knot surgery on K3 using different hyperbolic knots $K\subset S^3$. They all have vanishing SW invariants (and vanishing Bauer-Furuta invariants). But it is not known whether they are all diffeomorphic... if they are then there are some interesting properties of ASD Weyl curvature metrics on it!