What happens if electric field is not perpendicular to the magnetic field? What happens if the electric field is not perpendicular to the magnetic field for an electromagnetic wave?
What could be the possible consequences if such a case was found? 
Are there examples or demonstrations where the two fields are not perpendicular?
 A: In a word: nothing. 
It's perfectly easy to set up an electric field and a magnetic field at any arbitrary angle you might come up with - just set up a pair of electrodes and some Helmholtz coils at that angle and you're done. Similarly, if you want oscillating fields, you can set up monochromatic electric and magnetic fields with independent arbitrary polarization ellipses with only a modicum of work.
If you've got a single, monochromatic, linearly-polarized, plane-wave field, then yes, there is a standard argument that says that the electric field and magnetic field polarizations need to be orthogonal. However, this fails if you break any of those assumptions, particularly when you have non-plane-wave configurations or more than one plane wave.
The only actual constraints, in a vacuum, are the Maxwell equations $\nabla \times\mathbf E = -\frac{\partial\mathbf B}{\partial t}$ and  $\nabla \times\mathbf B = \mu_0\varepsilon_0 \frac{\partial\mathbf E}{\partial t}$ (plus, obviously, the two divergence laws). Everything else is secondary.
