$f:X to Y$ is a continuous map. Suppose it induces an isomorphism $f_*:H_*(X) to H_*(Y)$. Does that imply $f:X to Y$ is a homotopy equivalence?

In case when $Y={*}$ is a singleton your question can be stated as:

Suppose $X$ has all homology groups trivial. Is it contractible?

Unfortunately this is false. The double comb space is famously known to have all homotopy, homology and cohomology groups trivial, even though it is not contractible.

No; for example, there are non-contractible spaces that are acyclic (have trivial reduced homology), such as the Poincare homology sphere minus a point. You can take $X$ to be any such space and $Y$ to be a point.

However, this is true if $X$ and $Y$ are simply connected CW complexes, by a combination of the relative Hurewicz theorem and Whitehead's theorem. This is "Whitehead's theorem for homology."