Homology Whitehead theorem for non simply connected spaces

The answer is no. An easy example is provided in Allen Hatcher's Algebraic Topology, Example 4.35.

http://www.math.cornell.edu/~hatcher/AT/ATch4.pdf

There, a CW-complex $X$ is formed by attaching an appropriate $(n+1)$-cell to the wedge $S^1 \vee S^n$ of a circle and an $n$-sphere. The inclusion of the $1$-skeleton $S^1 \to X$ induces an isomorphism on integral homology and on homotopy groups $\pi_i$ for $i < n$ but not on $\pi_n$.

That example is also worked out as Problem 1 here:

http://www.home.uni-osnabrueck.de/mfrankland/Math527/Math527_HW10_sol.pdf

There are two finite 2-dimensional complexes $A, B$ which are not homotopy-equivalent but are homology-equivalent, i.e., there exists a continuous map $$ f: A\to B $$ inducing isomorphism of fundamental groups and homology groups. See the last paragraph (page 522) of this paper, the actual example is due to Dunwoody.

Edit 1: The homology equivalence part follows from Dyer's paper, see the link above (read the very last paragraph of his paper). Dyer's results are by no means obvious: He proves that equality of Euler characteristics plus one extra condition ($m=1$, whatever $m$ is) imply homology equivalence of 2-d complexes with isomorphic fundamental groups. Then Dyer verifies his conditions in the case of Dunwoody's example (the only nontrivial condition is $m=1$, since equality of Euler characteristics is clear).

Edit 2: The correct version of the "homological Whitehead's theorem" is indeed requires a cohomology isomorphisms with sheaf coefficients, see the discussion and reference here.

...it turns out there are non-trivial high dimensional smooth knots $S^n\subset S^{n+2}$ such that $\pi_1$ of the knot complement $X$ is infinite cyclic. The inclusion of the meridian $S^1 \subset X$ is a homology isomorphism, a $\pi_1$ isomorphism but is never a weak equivalence since by a result of Levine if it were then the knot would be trivial (we should assume $n \ge 3$ here).

John Klein @ https://mathoverflow.net/a/104534/1556