There are several ways that heat can be transmitted, but in this case we're talking about light, so it's helpful to talk about radiative heat, which is transmitted through infrared radiation that is invisible to our eye. In the case of a visible color, the fact that a material has color means that the material is absorbing some colors of light.

However, in general, the color of something might not be as significant as the general material of an object. (For example, is it metal, is it plastic, is it wood?) In this case, a black styrofoam cup would keep a liquid hot much longer than a white metal cup.

However, if everything else were the same we could look at very small differences in temperature, and if the color of a cup led to the cup absorbing less (that is, reflecting more) infrared light from the water, it should keep the water longer for a little longer. This would be a good opportunity to do an experiment!