No, this cannot be true if $f$ is just $C^\infty$. Let $u=e^{(\Delta+f)t}u_0=e^{\Delta t}v_0$. Then, by well known properties of the heat equation, $u$ and $u_t$ are spatially analytic for $t>0$. But this implies that $f$ is also analytic everywhere where $u\neq 0$.