If C and D are irreducible, affine varieties over an algebraically closed field, and I form the product variety CxD, is the projection morphism from CxD to C necessarily an open map? That is, is the projection of each Zariski open subset of CxD necessarily Zariski open in C?

Yes. More generally, flat morphisms locally of finite presentation are universally open (EGA IV_{2}, Théorème 2.4.6). 

