Navier-Stokes is an equation, not a theorem, so there is no such thing as "a proof of Navier-Stokes". The equation is a partial-differential-equation model of viscous fluid flow. Its correctness has never been in doubt: we know it cannot possibly be an exact description of real fluid flow, that it's a pretty good approximate description, and that there exist well-behaved solutions for a number of initial conditions.

What OpenAI purports to have proven, as I understand it, is that certain initial conditions to that equation, plus "forcing" over time (which could be a literal force acting on the fluid such as stirring it with a spoon or some other extrinsic effect) only have finite (and therefore physically plausible) solutions for a finite period of time, after which singularities appear, with the velocity or pressure of some of the fluid approaching infinity as you approach the finite time limit.

This is a result that Terry Tao conjectured in 02014, but without the forcing: http://arxiv.org/abs/1402.0290

I think we can be pretty confident that the L∃∀N proof is really about Navier-Stokes. The question is whether what it says about Navier-Stokes is what we think it says.