You could start by converting the image to grayscale and compareing that to the original image. If the image is already black and white, you'd expect the two images to be very similar; if it's full color, you'd expect them to be different.

Clarification: Given the two comments below, I think I wasn't clear enough here. Certainly, the grayscale version of a B&W image stored as RGB will likely not be exactly the same as the original image, but it should be pretty close. Subtracting the grayscale version from the original will yield a dark frame with a faint ghost of the original image. You could threshold this image to get a completely black frame. A full color image, on the other hand, will be significantly different from its grayscale counterpart, and subtracting the latter from the former will give an image that still has a lot of color even after thresholding (exactly what colors depends on how the grayscale conversion is done).

Another option: Thinking about the problem some more, another option is to convert the image to a neutral white point and then compare the red, green, and blue channels. In a color image, these will have significant differences; in a black and white image, they should be very similar. Again, they might not be exactly the same, depending on how the image was acquired in the first place, but the differences should be small enough that they can be easily filtered out. This is effectively the same idea as the one above, but the approach is perhaps more straightforward.