Point counting $K_0(Var/\mathbb{F}_q) \to \mathbb{Z}$ induced by $[X] \to \#X(\mathbb{F}_{q^e})$ is a ring homomorphism, so
$$[X] + [Y] = 0$$
would imply $\#X(\mathbb{F}_{q^e})  +\#Y(\mathbb{F}_{q^e}) = 0$. And this can only happen if $X$ and $Y$ are the empty varieties.


EDIT: The general idea in such a situation is to use the characterization of $K_0(Var/k)$ as the ring of universal Euler characteristics. In all of the beautiful comments mentioned here what is going on is on finds a function $Var/k \to R$ ($R$ a ring) which behaves as an abstract Euler characteristic. Then this factors through the Grothendieck ring of varieties.