Assumption: $X$ is a connected CW complex, and $H_{\ast}(X;\mathbb{Z})=\bigoplus_{n \geq 0} H_n (X; \mathbb{Z})$ is finitely generated.
Question: does there exist a finite CW complex $Y$ and a map $f:Y \to X$ inducing an isomorphism in integral homology?
Remarks:
The answer is almost trivially ''yes'' if $\pi_1 (X)=1$.
P. Vogel gave a positive answer to my question (''Un Theoreme de Hurewicz homologique'') if $H_1 (X; \mathbb{Z})=0$ and if $\pi_1 (X)$ satisfies an additional finiteness hypothesis (it has to be ''locally perfect'', i.e. each element is contained in a finitely generated perfect subgroup; a stronger condition that being perfect).
I am happy about a positive answer even under the restrictive assumption $H_1 (X; \mathbb{Z})=0$.
By the Kan-Thurston theorem, one might assume that $X=BG$ for a discrete group.
It is easy to produce an $f:Y \to X$, $Y$ finite, such that $f$ is surjective in homology: $X$ is the union of all its finite subcomplexes, and each homology class of $X$ is supported on one of them. Take a finite generating system for $H_{\ast}(X)$, pick a finite subcomplex for each generator, and take the union of these subcomplexes. It is also no big deal to achieve, by attaching cells to $X$, that $f$ is injective on $H_0$
and $H_1$ as well.
EDIT: the last part of the last sentence is wrong; thus I deleted my answer.