Concerning the analogous question for cohomology: There is a homeomorphism from the Klein bottle to itself that induces the identity map in integral cohomology but not in mod $2$ cohomology.
This example is Spanier-Whitehead dual to the $S^2\vee \mathbb RP^2$ variant of Sam's example.
For an example of a map inducing the identity in both integral homology and in integral cohomology, but not in mod $2$ (co)homology, you need to use a homology group that is not finitely generated.
In Sam's example (map from $\mathbb RP^2\vee \Sigma \mathbb RP^2$ to itself) the map is zero on both integral homology and integral cohomology.
