What happens depends strongly on the software to compute the solution, but in general if you don't have a proof of existence and uniqueness as well as a proof of convergence to the unique solution, you are on very shaky ground.
Let's consider $y' = f$ with $y(0) = 0$. This can be solved by an ODE solver, under some assumptions on $f$ (say, continuity). Now let's say $f$ is a delta function at a random location in the domain. A numerical solver will never find it, and the answer will always be wrong.
The next situation is non-uniqueness. I wrote a few solvers for a PDE which had many solutions. What we attempted was to compute them all, and then select a subset of the solutions based on some property we thought was relevant. (To be fair, what actually happened is we got into fights about what was relevant, and eventually we just modeled the phenomenon with a different PDE that did have a unique solution. I can honestly say that non-uniqueness was one of the biggest headaches I've had solving PDEs.)
I've never attempted to solve an equation which known to not have a solution, but I cannot imagine it turning out well. The best you can hope for is an error message, but you probably won't get it. My suspicion is that your numerical method will compute the solution to an equation which is in some sense "close" to the original, and then you'll be in a truly devastating situation where you have a bunch of garbage which you think is a solution.
Hopefully someone else can offer insight into Navier-Stokes; I've never worked on it so I can't help you there.