Is tensor rank is in VNP?
The collection of tensors of a given rank, or even of tensors with rank at most $k$ is not a (Zariski-)closed set, so it cannot be described as the vanishing locus of any set of polynomials, regardless of their complexity. (However, over finite fields tensor-rank is $NP$-complete and over $\mathbb{Q}$ it is $NP$-hard but not known to be in $NP$. But these are the usual Boolean classes, not the Valiant analogues.)

The closure of the the set of tensors of rank at most $k$ is the set of tensors of border-rank at most $k$. Call a set of polynomials whose vanishing locus is the set of tensors of border-rank at most $k$ a system of (set-theoretic) defining equations for border rank at most $k$. Such defining equations are known for small $k$, but for most $k$ finding such defining equations is a long-standing open problem, related the border-rank and multiplicative complexity of matrix multiplication.

See Landsberg's Bulletin article Geometry and the complexity of matrix multiplication for an introduction and some references, and see Landsberg's recent book Tensors: Geometry and Applications (freely available introduction) for all that is known about defining equations for border-rank.