The Dawn spacecraft is carrying 3 electromagnetic instruments; the Framing Camera (FC), the Gamma Ray and Neutron Detector (GRND), and the Visible and Infrared Mapping Spectrometer (VIMS). Your question is presumably about colour images in the visual spectrum and not false-color images of wavelengths not visible to humans, so let's focus on the Framing Camera.
Ceres as seen by the Dawn orbiter (Courtesy NASA)
According to NASA:
The Framing Camera is designed to acquire detailed optical images for
scientific purposes as well as for navigation in the vicinities of
Vesta and Ceres. Dawn carries two identical and physically separate
cameras for redundancy, each with its own optics, electronics and
structure. Each camera is equipped with an f/7.9 refractive optical
system with a focal length of 150mm and can use 7 color filters.
The camera on the Dawn orbiter, and most cameras in space, do not have the kind of image sensor a regular digital camera has. In this fascinating article on the Planetary Society website, Emily Lakdawalla writes:
Space cameras usually have detectors that can detect light across
slightly more of the electromagnetic spectrum than human eyes can see,
from near-ultraviolet to near-infrared wavelengths. More importantly,
while the pictures that they take are often pretty, they are not
intended to be for snapshots; they are sophisticated, precise
scientific measuring devices.
The FC is more of a 'photon measuring device' than a conventional camera. It measures the numbers of photons that land on each pixel of the sensor, not their wavelengths. However, it can perceive photons of different colours using filters.
A filter is like a gel on a theater light. It turns white light into
colored light. It does this by limiting the wavelengths that can pass
through the filter. Dawn's Framing Camera has eight different filters
fairly evenly spaced across the part of the spectrum that its CCD is
capable of detecting.
A graph showing the different wavelengths of visual light each filter (numbered F1-F8) allows through. (Courtesy CERN)
The image sensors in ordinary consumer cameras here on Earth actually also record light according to its intensity in three separate colours and then process that information to create a full-colour image. Different parts of the sensor are designed to respond only to certain wavelengths of light - some in red, some in green, some in blue. There are no sensors that can detect what wavelength of light is hitting them, they only count how many photons hit. One way or another, all cameras combine information gathered separately about light intensity at different wavelengths, and then combine that into a colour image. In fact, your eyes do this too.
Essentially, Dawn takes 'black-and-white' photographs at a very high resolution with each filter. Hence around eight images are created, each showing the raw photon-measurement in that respective section of the visual spectrum. These images can then be used to pretty accurately colourise the surface of Ceres and produce 'colour images'. However these would still be classified as 'false colour' pictures.
Why haven't any colour pictures of Ceres been released by Dawn? The answer is that they simply haven't been made yet. You can access the latest raw images from the Framing Camera here. NASA will presumably create colour pictures from the raw data in the future, and so could I, and so does @TildalWave. The colour these sorts of images are created on Earth, by people, not by the probe itself. There's no real reason to colourise an image other than beauty, the scientific data is gathered from the RAW file itself.
CERN document on the Framing Camera.
Planetary Society article on the Framing Camera.
Planetary Society article on the Dawn mission as a whole.