Take $n = 2$,$r = 0$ and $M = \mathbb R^2 - \{0,1\}$ and $N = \mathbb R^2 - \{0\}$ and $M \to N$ to be the natural inclusion.
Then on fundamental groups we have the surjection $F_2 \to \mathbb Z$ taking one generator to $1$ and the other to $0$. The kernel is a nontrivial subgroup of $F_2$, hence free and of cohomlogical dimension $1 \neq 0$.
(Note that if $M,N$ were compact, then by Ehresmann the fiber of $M \to N$ would be a classifying space for the kernel-- hence the kernel would indeed have the desired cohomological dimension.)