Which homomorphism $\pi_1(M) \to \pi_1(N)$ can be realized as induced homomorphism of a continuous map $f:M\to N$?
If $M$ is a one-dimensional Peano continuum and $N$ is a one-dimensional metric space, then the homomorphism is induced by a continuous map $f:M\to N$ up to a basepoint change. This is a result of Katsuya Eda.
The same holds if $M$ and $N$ are planar Peano continua (Greg Conner and Curtis Kent).
These cases do not assume local contractability and are quite non-trivial. The one-dimensional case can be used to show one-dimensional Peano continua with isomorphic fundamental groups are homotopy equivalent.
I don't think there is a simple general answer to this question (I'm assuming here that $M$ and $N$ are reasonable topological spaces like CW complexes). If $N$ is a $K(\pi,1)$, then this is possible. One may map the 2-skeleton of $M$ to $N$ using the homomorphism, and then map the higher skeleta in using the fact that maps of higher dimensional spheres into $N$ are homotopically trivial. This also works if the higher homotopy groups $\pi_k(N)$ vanish for $2\leq k\leq dim(M)-1$.