The boundary divisor $\Delta_0$ has intersection number $0$ with every $F$-curve of the form $\overline{\mathcal{M}}_{0,4}$, yet has nonzero intersection number with every $F$-curve of the form $\overline{\mathcal{M}}_{1,1}$.  

<B>Edit.</B>  Also, since $\lambda$ is the pullback of a divisor class from the Satake compactification of $\mathcal{A}_g$, also $\lambda$ has intersection number $0$ with every $F$-curve of the form $\overline{\mathcal{M}}_{0,4}$, yet has nonzero intersection number with every $F$-curve of the form $\overline{\mathcal{M}}_{1,1}$.