$\def\RR{\mathbb{R}}\def\Hom{\mathrm{Hom}}$This seems too simple, so please tell me what I'm missing. We first prove the corresponding result in homology. Let $C_i(X)$ be the group of singular $i$-chains in $X$. Then $C_i(M) = \lim_{j \to \infty} C_i(M_j)$ (because $i$-simplices are compact, so the image of any finite number of them must lie in some $M_j$). Homology commutes with direct limits, so we deduce that $H_i(M) = \lim_{j \to \infty} H_i(M_j)$.
Now, the universal coefficient theorem tells us that $H^i(M, \RR) = \Hom(H_i(M), \RR)$ and the same for $M_j$. So the question is whether $\Hom( \ , \RR)$ turns direct limits into inverse limits, and the answer is yes.
Both the universal coefficient theorem, and the identification of singular and de Rham cohomology, work for general manifolds.
I also have a direct proof of Mittag-Leffler for $H^q(M_0) \leftarrow H^q(M_1) \leftarrow \cdots$.
Lemma Let $M$ be a manifold (PL or better) and let $K$ be a compact subset of $M$. Then $H^q(M) \to H^q(K)$ has finite dimensional image.
Proof Take a PL triangulation of $M$. For each face $\sigma$ of the triangulation, let $U_{\sigma}$ be the union of the relative interiors of those faces $\tau$ containing $\sigma$. So the $U_{\sigma}$ form an open cover of $M$.
As $K$ is compact, it can be covered by finitely many $U_{\sigma}$. Let $N$ be the union of the closures of those $U_{\sigma}$. So $K \subset N$ and $H^q(M) \to H^q(K)$ factors through $H^q(N)$. But $N$ is a finite simplicial complex, so $H^q(N)$ is finite dimensional. $\square$
Remark Actually, the union of the $U_{\sigma}$ also has finite cohomology, without taking the closure, but I didn't see a one line way of saying it.
Now, let $M_0 \subset M_1 \subset M_2 \subset \cdots$ be an ascending chain of manifolds, with the closure $\overline{M_i}$ (in $M_{i+1}$) compact. Then $H^q(M_{i+1}) \to H^q(M_i)$ factors through $H^q(\overline{M}_i)$, so it has finite dimensional image. For each $j \geq i+1$, the image of $H^q(M_j)$ lies in that finite dimensional image. So the Mittag-Leffler condition holds and $\lim_{\infty \leftarrow j}^1 H^q(M_j)=0$.