Hmmm...I'm sorry I didn't notice this post earlier.
Yes, finches can be color-fed, but not all finches will show an effect in their feathers. What color-feeding is is adding substances (natural or synthetic) to the food, which the bird uses to enhance plumage coloration. Typically, there are two colors that are affected -- yellows and reds -- but not all finches with these colors produce them from dietary sources. For example, Gouldian finches can have red faces, but this red does not get enhanced by feeding red color-food. On the other hand, Bullfinches have reddish coloration to their bodies, and this IS affected by color-food. For species like Bullfinches, diets low in reddish-pigmented food will result in birds with little or no red coloration.
In general, a way to tell if color-feeding will have an effect is to look at the colored areas on the birds. Those areas that are sharply red or yellow, with no "bleeding" into other areas, are usually NOT dependent upon color-feeding (like the red masks of Gouldian finches). If, on the other hand, the color has a more suffused appearance, with intensity varying between individuals, then it's likely that color feeding WILL have an effect.
Additionally, birds must have the genes to express the color. If you feed canthaxanthin (the principle ingredient in red color-food) to society finches, you probably won't get a red society finch. Before breeders hybridized the Venezuelan Hooded Siskin with canaries, all the color food in the world would get (at best) a faded orange-yellow color. It was only when genes for producing red pigments derived from dietary sources were transferred to the canary that we were able to have Red-Factor canaries.
And one last thing -- in species which respond to color-feeding, the effect can be achieved only when they are color-fed while feathers are growing. This means starting just before the molt begins (at the sign of the first feather dropped) and continuing until the last of the new feathers has finished growing. Feeding color-food while a bird is not in a stage of active feather growth won't affect their coloration.
Long-story-short -- yes, they can, but it will work only if 1) the species has coloration that's influenced by diet, and 2) the birds are fed while feathers are actively growing.
So.....about what kind of finches are you asking?