Let $\tilde X$ be the blow up of $X\subset \mathbb{CP^{n+1}}$ along some closed subscheme $C\subset X$, $f : \tilde X\to X$ be the blow down map, and $f_*:H_k(\tilde X)\to H_k(X)$ be the induced map on homology groups. Is it true that $f_*$ is injective?
Edit
Should add the condition that $X$ is of dimension $n$ and $C$ is of dimension $n-2$, and only consider the homology group of degree $n-2$.
No!
Example: blowing up a point in $\mathbb{CP}^2$ gives $\widetilde{\mathbb{CP}^2}=\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ so its $H_2$ is $\mathbb{Z}^2$, generated by the exceptional divisor $E$ and the $\mathbb{CP}^1$ at infinity, but $H^2(\mathbb{CP}^2)=\mathbb{Z}$ is generated by the $\mathbb{CP}^1$ at infinity.so $f_*\colon H_2(\widetilde{\mathbb{CP}^2})\to H_2(\mathbb{CP}^2)$ cannot be injective. In fact $f$ shrinks the exceptional divisor $E$ to a point so $f_*[E]=0$ in $H_2$.
Maybe you are thinking about the map induced by the strict transform?