As far as I know the mass of an observed object increases as it approaches the speed of light.
Is it possible that the excess mass called "dark matter" is due to relativistic dust?
Surely, stars cannot account for that excess mass since we can measure their speeds and they are not relativistic. However, can we observe and measure the velocity of dust in the galaxy? Maybe the dust could account for missing matter...
There are several reasons this model fails, including
A significant portion of the dark matter is known to be gravitationally bound to galaxies and relativistic velocities are far about the escape velocity and the stuff you propose would not remain bound.
The stuff you propose would be hitting the Earth, the sun, the other planets and we'd see it. We do see relativistic material and call it "cosmic rays", but the mass is a small fraction of the baryonic matter density.
The stuff you propose would hit the gas and dust of nebulae and cause their temperature to rise in a way that we could observe. This is not seen.
In addition to the observations described in dmckee's answer, there is a separate thread of cosmological evidence that shows that dark matter is nonbaryonic.
Big Bang nucleosynthesis gives the wrong relative abundances of H, He, and deuterium if there is so nonbaryonic matter. In the BBN period, dust couldn't exist, since all matter was completely ionized.
The existence of nonbaryonic matter is also required in order to reconcile the observed density of galaxies with the observed strength of the CMB fluctuations.
There's a 2012 review paper by Roos, http://arxiv.org/abs/1208.3662 .
