What an interesting question! I can honestly say I haven't heard of this before or even really considered it before! In order to solve this, we first need to find the mass of the Earth and the mass of an average mole. The mass of the planet is approximately 5.98 x 10^24 kg. As best I can tell, an adult mole weighs between 34 and 170 grams, depending on the particular species. So the average of the two ends of the weight scale would be (34+170)/2=102 grams which is the same as 0.102 kg, to get both measurements in the same units.
We know that, by definition, a mole of any substance is 6.022 x 10^23 units of that substance. This is also called Avogadro's number. So a mole of moles would be equal to 6.022 x 10^23 moles. If each mole weighed the average of 0.102 kg, then a mole of moles would weigh 6.022 x 10^23 * 0.102 kg = 6.14 x 10^22 kg. Since the mass of the Earth is 5.98 x 10^24 kg, then the mass of a mole of moles would actually be less than the mass of the Earth. The exact measurement is:
6.14 x 10^22 kg/5.98 x 10^24 kg = 0.0103
Multiply this by 100 to get 1.03%. So a mole of moles would actually only be about 1.03% the mass of the Earth.