2 planes in same location - preferred render

When 2 planes are exactly in the same location, blender and blend4web will render "jiggy".
Do I have to clean up my model or is there a way to tell renderer that first object must be rendered and the other one ignored in that place?

Yeah, it's better to clean up the model. To do it more easily you can use Boolean modifier

It's called Z-fighting and appears when two polygons are very close to each other and they start "fighting" - trying to be rendered first

You could also try adjusting the near and far clip plane on the camera, setting it so that you only render what is supposed to be seen in the scene. If the faces are at the same location then z-fight will appear anyway.

That's weird. Sometimes the z-fighting does not happen, even when the planes are overlapping exactly.
And sometimes it happens, when 2 planes are very close but zoom is very out.
Here is simple example, where z-fight does not happen:
cubes.html
cubes.blend

That's weird. Sometimes the z-fighting does not happen, even when the planes are overlapping exactly.
And sometimes it happens, when 2 planes are very close but zoom is very out.

Z-buffer is non-linear and have the best precision near the camera. Its precision also depends on device, e.g. on some mobile devices z-fighting may occur even if it does not on desktop.

Here is simple example, where z-fight does not happen:

Well, this is probably because both cubes have exactly same materials and are lit in the exact same manner - so you cannot see the artefacts. I tried changing materials to make sure.