The proof goes roughly as follows: Let $u$ be exact solution and $u_h$ discrete solution. Moreover, let $e=u-u_h$ and $\pi_h$ be the interpolation operator from $H^1$ to the discrete finite element space. Facts used in proving the a posteriori error estimate are the following:
- Coercivity of the bilinear form.
- Galerkin orthogonality.
- Problem statement and elementwise integration by parts.
- Cauchy-Schwarz for inner products and for sums.
- Interpolation estimates $\|e-\pi_h e\|_{0,K} \lesssim h_K |e|_{1,K}$ and $\|e-\pi_h e\|_{0,E} \lesssim h_E^{1/2} |e|_{1,K}$ and the definition of $\|\cdot\|_1$.
$$\begin{align*}
\|u-u_h\|_1^2&\lesssim (\nabla(u-u_h),\nabla(u-u_h))\\
&=(\nabla(u-u_h),\nabla(e-\pi_h e))\\
&=(f,e-\pi_h e)+\sum_{K}(\Delta u_h, e-\pi_h e)_K-\sum_{E}([[\nabla u_h\cdot n]],e-\pi_h e)_E\\
&\lesssim \big(\sum_{K} h_K^2 \|\Delta u_h+f\|_{0,K}^2\big)^{1/2} \big(\sum_{K} h_K^{-2}\|e-\pi_h e\|_{0,K}^2\big)^{1/2}\\
&\phantom{=}+\big(\sum_{E} h_E \|[[\nabla u_h\cdot n]]\|_{0,E}^2\big)^{1/2} \big(\sum_{E} h_E^{-1}\|e-\pi_h e\|_{0,E}^2\big)^{1/2}\\
&\lesssim \Big(\big(\sum_{K} h_K^2 \|\Delta u_h+f\|_{0,K}^2\big)^{1/2}+\big(\sum_{E} h_E \|[[\nabla u_h\cdot n]]\|_{0,E}^2\big)^{1/2} \Big)\|u-u_h\|_1
\end{align*}$$
Finally you divide by $\|u-u_h\|_1$ to get the bound.
In particular, note that none of the steps are invalid for $u_h|_K \in P^1(K)$. Thus, $\Delta u_h|_K=0$ should work just fine.