Can you point out some references that deal with the obstacle problem for the heat equation?

$$(OP) \quad\begin{cases} \max\{\Delta u -\partial_t u, \varphi - u \} = 0 & \text{ in } (0,T)\times \mathbb{R}^n \\ u(0,\cdot) = \varphi(0,\cdot) & \text{ in } \mathbb{R}^n \end{cases}.$$

Works on elliptic obstacle problems appear to be much easier to find (see Wikipedia, for instance).