I know that we can use mathematical analysis techniques to prove if an IVP or BVP has a solution, is unique, and depends continuously on the boundary / initial values. For some PDE's, particularly non-linear pde's, it's very difficult, if not impossible, to prove well-posedness. Is there any kind of numerical technique to verify if a problem is well-posed or not?
In general, no. A numerical solution can sometimes be used as a rough measure to indicate whether boundary conditions are sufficient, to identify "floating" domains for example, but there are many cases in which discrete solutions give you downright misleading information about the continuum problem.
Advection-diffusion requires a boundary condition on all boundaries, but discrete systems can use no boundary condition at the outflow (not a homogeneous Neumann condition, I really mean no boundary condition). Not only that, it's more accurate than the discrete representation of the continuum boundary condition. See Papanastasiou, Malamataris, and Ellwood 1992 and Griffiths 1997 for details. A similar boundary condition is also important for slip on curved surfaces, see Behr 2004.
The "carbuncle phenomenon" afflicts certain methods for compressible flow. It is not very well understood, but seemingly robust numerical schemes may converge to spurious solutions. An example from Robinet et al. 2000
Spurious solutions to incompressible Navier-Stokes, within a laminar regime. A simple lid-driven cavity example is given in Schreiber and Keller 1983.
Systems of hyperbolic conservation laws with non-physical relative size of numerical dissipation. Some numerical dissipation is always required, but otherwise robust (e.g. Godunov) methods can systematically converge to incorrect results if numerical dissipation ends up being non-physical. A simple example is given in Mishra and Spinolo 2011 where the standard Godunov method converges to an incorrect result for 1D linearized shallow water. This presents itself in a deeper form in large eddy simulation. The eddy viscosity is a physical manifestation of subgrid scales, but if the (unavoidable) numerical dissipation is larger than the physical dissipation, the simulation can converge to systematically incorrect results. In practice, the subgrid closures for eddy viscosity are very important. This is a matter of taking a singular limit along the correct (physical) path.
Locking effects in elasticity or checkerboard modes in incompressible flow. These are due to choosing an unstable approximation space and are now very well-understood, at least for linear problems, but relying on a numerical solution to deduce well-posedness might lead you to conclude that the incompressible limit was ill-posed.