Let $n<m$ be natural numbers and consider the C*-algebras $M_n$ and $M_m$ of matrices. Suppose that $f\colon M_n\to M_m$ is a a completely positive (linear) map. Is it true that
$$f(M_n)\subseteq \mbox{span}\{e_{i,j}\colon i\in I, j\in J\},$$
where $I$ and $J$ are some $n$-element subsets of $\{1,\ldots,m\}$ consisting of consecutive integers, that is, are of the form $\{k+1\ldots, k+n\}$ for some $k<m$?
As Michael said, that's not true even for *-monomorphisms. For example, consider $\pi:M_2(\mathbb C)\to M_4(\mathbb C)$, given by $$ \pi:\begin{bmatrix}a&b \\ c&d\end{bmatrix}\to\begin{bmatrix}a&0&0&b\\0&0&0&0\\0&0&0&0\\c&0&0&d\end{bmatrix}. $$ This is a $*$-monomorphism, so it is in particular completely positive.