  Let $(M,J,\theta)$ be a complete pseudo-Hermitian manifold which satisfies
the CR sub-Laplacian comparison property. In this paper, we derive the local
subgradient estimates for positive solutions to the following nonlinear
subparabolic equation: \begin{equation*} u_t=\Delta_\mathrm{b}u+au\log u+bu,
\end{equation*} on $M\times [0,+\infty)$, where $a,b\in \mathbb{R}$. As a
application, we derive a priori estimate and a Harnack inequality for positive
solutions to the subelliptic equation $\Delta_\mathrm{b}u+au\log u=0$.
