To render vertex color information with mental ray you need to use the metalrayVertexColors node. When you do this you will have a white plane with some transparent patches. The tutor assigns white with alpha = 1 (solid) to the whole mesh just before he starts painting white on white but with the alpha = 0 (transparent). To get an idea of what you will see in the render when you use the vertexcolors node, switch your screen renderer to High Quality or Viewport 2.0.
To get around this you can try baking the existing color information of the texture material into the vertices. I'm note sure if you will loose the existing alpha information when you bake it. The down side is you will loose most of the detail as there are just not enough vertices.

Any particular reason you want to use vertex painting?