Example of an unstable map between finite complexes which is the identity on homotopy but not homotopic to the identity?

Pick a degree $1$ map $h: T^3 \to S^3$ from the $3$-torus to the sphere and define $$f: T^3 \times S^3 \to T^3 \times S^3; \; f(x,y):=(x, yh(x)).$$ This map induces the identity on homotopy groups, but not on homology.

If $X = S^1 \times \Bbb{CP}^\infty$, then there is a map $X \to X$ with the following property. For any space $Y$, $[Y,X] \cong H^1(Y) \times H^2(Y)$, and so the map $X \to X$ classifies the map $$ (a,b) \mapsto (a, a^2 + b). $$ This is clearly not homotopic to the identity self-map of $X$ because it's not the identity natural transformation.

The fact that this is the identity on homotopy groups follows by considering the case $Y = S^n$: in both of those cases, the map $H^1(S^n) \times H^2(S^n)$ given by $(a,b) \mapsto (a,a^2 + b)$ is the same as the identity map $(a,b) \mapsto (a,b)$. [Yes, I neglected the basepoints and they are important but they don't alter this outcome.]

Unfortunately this isn't a finite complex. It also falls into the category of being "$1+f$" for some $f$ which is zero on homotopy, as you suggest, but $f$ in this case is not really a phantom map on its own -- it doesn't restrict to the zero map on the finite subcomplexes.

George Cooke gave an example in Trans, AMS 237 (1978) 391-406. Define a map $h\colon S^n \times S^n \vee S^{2n} \to S^n \times S^n \vee S^{2n}$ by $$ \begin{cases} h|_{S^n \vee S^n \vee S^{2n}} = \mathrm{id} \\ h|_{2n-cell} \ \mathrm{wraps\ non-trivially\ around}\ S^{2n}. \end{cases} $$ This induces the identity on homotopy, but is non-trivial on homology. This in fact gives a homotopy action of $\mathbb{Z}$ on the space and Cooke showed how this could be replaced by a true action on a homotopically equivalent space. As far as the question about $\mathrm{Aut}(X)$, the paper by Pak and me in Topology and its Applications 52 (1993) 11-22 might be relevant. We constructed examples where the representation from components of the homeomorphism group to $\mathrm{Aut}(\pi_*(X))$ is not faithful. This involved interactions between the notions of simple space, the Gottlieb group, principal bundles over tori and the $h$-rank of a space. It all went back to the old question of Gottlieb: is there a finite simple complex with non-trivial fundamental group, but trivial Gottlieb group.