Are close maps homotopic?

This is true in the case that $N$ is compact (for example one can use the exponential map of an arbitrary Riemannian metric, together with the fact that its injectivity radius is positive to construct a homotopy), but not in general.

For a counter-example take $M = \mathbb R^2$ and $N =\mathbb R \times \{0\} \cup \mathrm{graph}(\frac 1x)$. Then for any $\epsilon >0$ we can find points $x,y$ in $N$ which lie at a distance at most $\epsilon$ to each other, but not in the same connected component. Hence the constant maps $f = x$, $g=y$ are not homotopic.

For a slightly less trivial example, consider $M = N = \mathbb R^2 \setminus \{0\}$ and $$f(x) = \delta \frac{x}{\vert x\vert}, \qquad g(x) = \delta$$

for sufficiently small $\delta$.