Of course, the answer to this question depends on what we mean by suspension. If we work with based spaces and take the reduced suspension, the answer seems to be NO:

Take $X = \mathbb N$ (a discrete countably infinite space) with basepoint $0 \in \mathbb N$; and $Y = \{1/n \ | n \in \mathbb N\} \cup \{0\}$ with basepoint $0$. Then the obvious bijection $X \rightarrow Y$ is a weak equivalence, but ($S$ is the reduced suspension) $S X$ is a wedge of circles, whereas $S Y$ are the Hawaiian earrings.

So my question is the following:

Is the statement true if we take the unreduced suspension instead?

Remark: If the spaces are connected, the statement seems to be true, no matter which suspension we take: This follows from the fact that the suspension of the given map is a homology equivalence between simply connected spaces, hence the Hurewicz theorem applies to conclude that its cofiber is contractible.