I have spent probably more time than I should attempting to verify this fact which is a question in Da Silva's "Lectures on Symplectic Geometry." I attempted to show it is $C^{\infty}(M)$-linear as follows:
"We only compute $N(v, fw)$ for $f \in C^{\infty}(M)$ at the point $x \in M$, and use the fact that at a given $x$, $J_x$ is linear map on tangent spaces and so it distributes over the tangent vectors we obtain by evaluating the vector fields at $x$. We see that:
\begin{align*} [Jv, Jfw]_x &= J_xv_x(J_xf(x)w_x) - J_xf(x)w_x(J_xv_x) \\ -J_x[Jv, fw]_x &= -v_x(f(x)w_x) + J_xf(x)w_x(J_xv_x) \\ -J_x[v, Jw]_x &= -J_xv_x(J_xf(x)w_x)-f(x)w_x(v_x) \\ [v, fw]_x &= v_x(f(x)w_x)-f(x)w_x(v_x) \end{align*}
Now combining like terms we obtain $-2f(x)w_x(v_x)$. By inspection we see that $N(v, w)$ produces the same result without the $f$, and therefore $fN(v, w) = N(v, fw)$. Since we have demonstrated that this is $C^{\infty}(M)$-multilinear, it must be a tensor."
I am currently feeling very uncomfortable working with the almost complex structure and the Lie bracket (I recently found out that $J$ does not distribute over $vw-wv$ since they are not guaranteed to be vector fields), and any corrections to errors in this proof would be much appreciated.
Thank you