If $M$ is a smooth $n$-manifold, the famous Whitney embedding theorems show that we can view $M$ as an embedded submanifold of some Euclidean space $\mathbb{R}^N$.
Does the Heine-Borel theorem still hold for subsets of $M$? I.e., is $S\subseteq M$ compact iff $S$ is closed in $M$, and bounded in the induced metric from $\mathbb{R}^n$?
No. For example, if $M=(0,1)$, then $M$ is closed in itself and embeds boundedly in $\Bbb{R}$, but is not compact.
If $M$ were embedded as a closed submanifold of $\Bbb{R}^N$, the result you want would follow immediately. But there's no guarantee that $M$ has a closed embedding in $\Bbb{R}^N$ just because it has an embedding in $\Bbb{R}^N$.