These are known as [*acyclic spaces*](https://en.wikipedia.org/wiki/Acyclic_space) (note that since $\tilde C_*(X)$ is a bounded below complex of projectives, it being contractible is equivalent to its homology being trivial).

There's an extensive literature about them, starting with this [Emanuel Farjoun's paper](http://www.math.huji.ac.il/~farjoun/acyclic%20spaces.pdf).

In general, yes $C_*(X)$ does forget some information about $X$: first of all it factors through the stable homotopy type $\Sigma^\infty_+X$ of $X$, and it forgets further information from there (the slogan is that it remembers only the "$\mathbb{Z}$-linear" information contained in $\Sigma^\infty_+X$). For example it is unable to distinguish between $\mathbb{CP}^2$ and $S^2\vee S^4$ (since the attaching map $\eta:S^3\to S^2$ is sent to a map homotopic to 0 by $C_*(-)$).

You can get much closer to reconstructing the full homotopy type of $X$, by remembering the coalgebra structure on $C_*(X)$ induced by the diagonal, although in general that is still not enough.