The statistical mechanical approach to Casimir problems for dielectrics separated by a vacuum gap turns out to be compact and effective. A central ingredient of this method is the effect of interacting fluctuating dipole moments of the polarizable particles. At arbitrary temperature the path-integral formulation of quantized particles, developed by Høye-Stell and others, is needed. At high temperature-the limit considered in the present paper-the classical theory is, however, sufficient. Our present theory is related to an idea put forward earlier by Jancovici and Samaj (2004), namely, to evaluate the Casimir force between parallel plates invoking an electronic plasma model and the Debye-Hückel theory for electrolytes. Their result was recently recovered by Høye (2008), using a related statistical mechanical method. In the present paper we generalize this by including a constant permittivity in the description. The present paper generalizes our earlier theory for parallel plates (1998), as well as for spherical dielectrics (2001). We also consider the Casimir force between a polarizable particle and a conductor with a small density of charges, finding agreement with the result recently derived by Pitaevskii (2008).