The effect is real and is called conservation of angular momentum. As the atmosphere warms, it expands slightly causing its moment of inertia to increase.
Now, the "quote", as written, is pure gibberish, but I suspect this is due more to the reporter being a mental midget in the area of science and physics than to the Doctor being wrong.
I don’t think there is any way that would increase the mass.
In any event, if you check the posters reply post to mine, you’ll see it’s just satire.