It's difficult to tell. Dreams are very hard to analyse scientifically since they can't be objectively measured, only self-reported. Dreams are notoriously difficult to recall after waking, so it's almost impossible to tell for certain.
There are some self-report studies which do assert that some proportion of dreams are in black and white, but this pattern is not seen in dream reports in various historical sources, so a plausible explanation is that black-and-white dreaming is caused mainly by exposure to black-and-white film media throughout the early 20th century, and that the percentage of dreams that are in black-and-white are skewed toward older populations who grew up with black-and-white films (though I don't personally know of any hard data on this point)
See this article by Schwitzgebel, 2001, who recounts several primary sources to this effect: http://philsci-archive.pitt.edu/267/1/dreamb&w010430.pdf
Schwitzgebel, E. (2002). Why did we think we dreamed in black and white?. Studies in History and Philosophy of Science Part A, 33(4), 649-660. PDF