No, because the space of tensors of order n with k-dimensional arguments is k^n-dimensional, while a complete n-partite graph with partitions of size k has only nk(n-1)k/2 edges, which doesn't give you enough coefficients to uniquely identify a tensor if n > 2.

You'd need a n-uniform n-partite hypergraph, where coefficients are assigned to n-tuples instead of pairs. So for a tensor of order 3, that would mean weighted triangles, which are somewhat hard to draw and everything gets less intuitive.