Invariants and orbits of $n$-tensors

Here is a start, suppose that $V_i$ is $\mathbb C^{k_i}$ (and restricting to $k_1,k_2,\dots,k_n, n\geq 2$). The tuples $(k_1,k_2,\dots,k_n)$ for which the action of $GL_{k_1}\times\cdots\times GL_{k_n}$ on $\mathbb{C}^{k_1}\otimes \cdots\otimes \mathbb{C}^{k_n}$ has only finitely many orbits are $(k,l),(2,2,k),(2,3,k)$, for positive integers $k,l$. This was proven in

V. G. Kac, "Some remarks on nilpotent orbits", J. Algebra 64 (1980), 190–213.

These orbits are classified in "Orbits and their closures in the spaces $\mathbb{C}^{k_1}\otimes \cdots\otimes \mathbb{C}^{k_r}$" by P.G. Parfenov (MR). This paper doesn't seem to be freely online in English, but Russian version is here, and I believe you can find a summary in section 5 here.

I don't think that's right, Bugs. Say that $V_1$ and $V_2$ are $d$-dimensional and $V_3=\mathbb C^2$. Then an element of $V_1\otimes V_2$ is a linear map $V_1^\star\to V_2$, generically an isomorphism; an element of $V_1\otimes V_2\otimes V_3$ is an ordered pair $(A,B)$ of these; the unordered $d$-tuple of eigenvalues of $B\circ A^{-1}$ is an invariant of the $GL(V_1)\times GL(V_2)$-action; and an element of $GL(V_3)$ will just perform some fractional linear transformation on all of these numbers, so that if $d\ge 4$ then there is a complex invariant here.