I read Cooperstock & Tieu (2006) and will discuss it a bit.
Before getting started, we should remember that galactic rotation curves are not the only evidence for dark matter. Gravitational lensing surveys and measurements of the cosmic microwave background both point to the existence of dark matter. And all three (rot. curves, lensing, CMB) agree on the approximate ratio of baryonic matter to dark matter. When a single model can explain multiple phenomena, that is strong evidence in favor of the model.
Background
The Newtonian gravity is a good model for motion around a point mass in the weak field, slow speed limit: $\frac{GM}{r}\sim v^2\ll 1$ (in units where $c=1$). Starting from the Schwarzschild metric, one can take the appropriate limit and get Newtonian gravity.
Newtonian gravity is a linear theory, so the gravitational potential due to multiple point masses is the same as the sum of the individual gravitational fields. This is because the field equation is a linear differential equation:
$$ \nabla^2 \varphi = 4\pi G \rho,$$
where $\varphi$ is the potential and $\rho$ is the mass density.
The usual method to determine the rotation curve of a galaxy with Newtonian gravity relies on a few assumptions: the gravitational field is weak, the galactic material is moving slowly. There's also a hidden assumption that non-linear effects from GR are small. This is true for 2-body dynamics, where you can track the appearance of non-linear terms in the post-Newtonian expansion.
A relativistic galactic model
Instead of modeling the galaxy as a collection of Newtonian point masses all orbiting their collective center of mass, Cooperstock & Tieu want to start with a fully relativistic, rotating cloud of dust. This is related to the van Stockum dust metric. Then they can take an appropriate limit of the relativistic system to get rotation curves. If the Newtonian assumption was right all along, then their method should give the same result as the Newtonian one. But they get something different.
The metric for their rotating dust contains non-zero, off-diagonal terms in $g_{t\phi}$. This is the same type of term that results in frame dragging in the Kerr metric. They define the important part of that off-diagonal term as $N$. If all of the terms containing $N$'s were suppressed by one of the small factors, $\frac{GM}{r}$ or $v^2$, then none of this would matter. To first order, Newtonian gravity would be fine. Cooperstock & Tieu say that that is not the case. The off-diagonal term $N$ matters.
The contribution of $N$ is non-linear. You can't just add up the gravitational contribution from each dust particle to get the answer using Newtonian gravity.
Did they make a mistake?
The second half of the paper attempts to address concerns from other scholars. There are a lot of concerns, and much of them are quite technical. Without digging into the whole thing in more detail I can't tell for sure who is right.
The most convincing criticism to me is that the authors didn't start from the true, fully relativistic metric, but jumped ahead to an already approximate, weak field state. Bratek et al (2006) claim that there is no Newtonian limit to a fully relativistic stationary, axisymmetric, and asymptotically flat spacetime of dust. So the whole endeavor is flawed from the start.
Another common criticism is that their solution has a discontinuity in $\frac{\partial N}{\partial z}$ at $z=0$. This leads to a singular surface, basically a thin sheet of mass, rendering the whole thing unphysical. The mass distribution isn't really just dust, there's some exotic matter in the $z=0$ plane. Cooperstock & Tieu argue this is a mathematical artifact, not a real problem.
One way to fix this singular surface is to force $\frac{\partial N}{\partial z} = 0$ at $z=0$, but this results in a loss of asymptotic flatness. This points me right back to the Bratek paper...
