I know the problem is especially of interest to mathematicians, but I was wondering if a solution to the problem would have any practical consequences.
Upon request: this is the official problem description of the aforementioned problem.
This is difficult to answer, because the answer depends on whether the answer is positive or negative. Although most people, including myself, expect a positive answer, a negative answer is actually possible, unlike say, for the question of P!=NP, or the well-definedness/mass-gap of gauge theory where we are 100% sure that we know the answer already (in the scientific, not in the mathematical sense).
If the answer is positive, if all Navier Stokes flows are smooth, then the proof will probably be of little practical consequence. It will need to provide a new regularity technique in differential equations, so it will probably be of great use in mathematics, but it will not be surprising to physics, who already expect smoothness from the known approximate statistical falloff of turbulence.
At short distances, the powerlaw falloff in turbulence turns into an exponential falloff, in the dissipative regime, so that the high k modes are suppressed by exponentials in k, and this implies smoothness. So smoothness is the expected behavior. This doesn't provide a proof, because the k-space analysis is too gross-- it is over the entire system. To prove the smoothness locally, my opinion is that you need a wavelet version of the argument.
On the other hand, if there are blow ups in Navier Stokes in finite time, these are going to be strange configurations with very little dissipation which reproduce themselves in finite time at smaller scales. Such a solution might be useful for something conceivably, because it might allow you to find short distance hot-spots in a turbulent fluid, where there are near-atomic-scale velocity gradients, and this could concievably be useful for something (although I can't imagine what). Also, such solution techniques would probably be useful for other differential equations to find scaling type solitons.
So it is hard to say without knowing from which direction the answer will come.