The Poincaré inequality need not hold in this case.  The region where the function is near zero might be too small to force the integral of the gradient to be large enough to control the integral of the function.

For an explicit counterexample, let $$\Omega = \{(x,y) \in \mathbb{R}^2 : 0 < x < 1, 0 < y < x^2\}$$ be the region under the graph of a parabola, and take $p=2$.  For $\epsilon > 0$, let $$u_\epsilon(x,y) = \begin{cases} x/\epsilon, & x < \epsilon \\\ 1, & x \ge \epsilon \end{cases}.$$  $0$ is in the essential range of each $u_\epsilon$, but one can easily verify $||\nabla u_\epsilon||_2^2 = \epsilon/3 \to 0$ as $\epsilon \to 0$, whereas $||u_\epsilon||_2^2 \to |\Omega| = 1/3$.