As per Theo Johnson-Freyd's request, I'm converting my comment to an answer.

Larsen-Lunts show that if $k$ is algebraically closed of characteristic zero, then there is a natural isomorphisms $$K_0(\text{Var}_k)/(\mathbb{L})\overset{\sim}{\longrightarrow} \mathbb{Z}[SB],$$
where $SB$ is the monoid of stable birational equivalence classes of smooth birational varieties and $\mathbb{L}=[\mathbb{A}^1]$. (There is a way of making sense of this if $k$ is not algebraically closed, but the target will no longer be a monoid ring, so the argument I am making will break.)

$\mathbb{Z}[SB]$ is manifestly torsion-free; thus if $nX=0$ for $n\in \mathbb{Z}$ and $X\in K_0(\text{Var}_k)$, we must have $X\in (\mathbb{L})$. Thus either we would have an example where $\mathbb{L}$ is a zero divisor, or a proof that $$\bigcap_n (\mathbb{L}^n)\neq 0.$$ $\mathbb{L}$ was recently shown to be a zero divisor by Borisov, but this was open for a long time; as far as I know, the question of whether or not $$\bigcap_n (\mathbb{L}^n)=0$$ is open as well (I've thought about it a bit, because I used the $\mathbb{L}$-adic topology on $K_0(\text{Var}_k)$ in my paper "Symmetric Powers do not Stabilize," and I think it's quite hard). So this question is almost certain to be open for $k$ algebraically closed of characteristic zero.