Let $\mathcal A$ be an Abelian category, $X_1$ a positive projective complex of $\mathcal A$, $X_2$ a complex of $\mathcal A$ and $f,g:X_1\to X_2$ two chain maps.
Suppose $H_n(f)=H_n(g)$ holds for any $n\in\mathbb Z$, can we deduce that $f$ and $g$ are chain homotopic?
No: For example, take $X_1=P$ to be a projective resolution of an object $A\in{\mathscr A}$, and $X_2=\Sigma^n B$ to be a stalk complex on another object $B\in{\mathscr A}$. In this case, then chain homotopy classes $[X_1,X_2]=[P,\Sigma^n B]$ correspond to extension classes in $\text{Ext}_{\mathscr A}^n(A,B)$, of which their might be several. On the other hand, if $n\neq 0$, any such map $P\to\Sigma^n B$ induces the zero map in cohomology, because cohomologies have disjoint support.