We prove gradient bounds for the Markov semigroup of the dynamic $\varphi^4_2$-model on a torus of fixed size $L>0$. For sufficiently large mass $m>0$ these estimates imply exponential contraction of the Markov semigroup. Our method is based on pathwise estimates of the linearized equation. To compensate the lack of exponential integrability of the stochastic drivers we use a stopping time argument and the strong Markov property in the spirit of Cass--Litterer--Lyons. Following the classical approach of Bakry-Émery, as a corollary we prove a Poincaré/spectral gap inequality for the $\varphi^4_2$-measure of sufficiently large mass $m>0$ with almost optimal carré du champ.