We study how quenched impurities affect the surface diffusion and ordering of strongly interacting adsorbate atoms on surfaces. To this end, we carry out Monte Carlo simulations for a lattice-gas model of O/W(110), including small concentrations of immobile impurities which block their adsorption sites. We examine the behavior of the diffusion coefficients and order parameters as a function of coverage corresponding to various ordered phases at low temperatures. The effects of impurities are examined under both equilibrium and nonequilibrium conditions, and the results are compared to recent studies on a completely clean surface. We find that even minute impurity concentrations affect the diffusion behavior considerably in equilibrium. The effects are strongest in ordered phases and close to phase boundaries, where quenched impurities lead to a reduction of order, which in turn leads to significant changes in the collective diffusion and phase behavior. As the impurity concentration is increased to a level of a few percent of the total surface area, the reduction in order becomes particularly prominent at high coverages. Further studies under nonequilibrium conditions reveal that nonequilibrium effects are strong in the absence of impurities, while for surfaces covered by impurities the nonequilibrium effects are relatively weaker.