Due to the coupling between the plasma membrane and the actin cytoskeleton, membrane molecules such as receptor proteins can become immobilized by binding to cytoskeletal structures. We investigate the effect of immobile membrane molecules on the diffusion of mobile ones by modeling the membrane as a two-dimensional (2D) fluid composed of hard particles and performing event-driven molecular dynamics simulations at a particle density where the system is in an isotropic liquid state. We show that the diffusion coefficient sharply decreases with increasing immobile fraction, dropping by a factor of ∼ 3 as the fraction of immobile particles increases from 0 to 0.1, in a system-size dependent manner. By combining our results with earlier calculations, we estimate that a factor-of-∼ 20 reduction in diffusion coefficients in live cell membranes, a puzzling finding in cell biology, can be accounted for when less than ∼ 22% of the particles in our model system is immobilized. Furthermore, we investigate the effects of confinement induced by a correlated distribution of immobile particles by calculating the distribution of the time it takes for particles to escape from a corral. In the regime where the particles can always escape from the corral, it is found that the escape times follow an exponential distribution, and the mean escape time grows exponentially with the density of obstacles at the corral boundary, increasing by a factor of 3-5 when immobile particles cover 50% of the boundary, and is approximately proportional to the area of the corral. We believe that our findings will be useful in interpreting (1) single molecule observations of membrane molecules and (2) results of particle based simulations that explore the effect of fluid dynamics on molecular transport in a 2D fluid.