I can imagine a map $f: X\to Y$ which is a homotopy equivalence of unpointed spaces, but which is not a homotopy equivalence of pointed spaces, no matter what basepoint is chosen. That being the case, I don't see why $f$ would have to be a weak homotopy equivalence.
More detail: by choosing $x\in X$, and its image $y\in Y$ as basepoints, we get a pointed map and an induced map on homotopy groups. To be a weak homotopy equivalence, this needs to be an isomorphism (and one point is as good as any point if $X$ is path connected). But this (hypothetical) pointed map is not invertible in the homotopy category of pointed spaces, so why should it induce an isomorphism?