Besides, to whom would one sell, say, Michelangelos Pieta? George Soros? So he could put it in his private collection? To whom could one sell the Sistine Chapel? The Saudis? So they could carve it to pieces and put the fragments on the world market? Or blow it up like the Bamiyan Buddhas? (Link)
Second, the great art functions not as a financial resource, but as a net drain on the Vatican's budget, as they can hardly afford even now to maintain and preserve them. You might have noticed that thirty years ago, the Sistine repair/restoration work was put in the hands of the Nippon Television Network Corporation (here's the story on the NTNC deal (Link) because the Vatican itself could not afford the sophisticated and painstaking technical process. Nippon TV undertook the restoration in exchange for exclusive photography and videotaping rights.
The story covered here on a Free Republic thread http://www.freerepublic.com/focus/f-religion/2694740/posts about how a huge library of rare manuscripts and books from ancient Eastern/Oriental Christianity is in imminent danger of falling into ruin because the Vatican can't afford to store them in the kind of temperature-and-humidity controlled, environmentally exacting conditions necessary for their preservation.
The whole thing is very briefly put in perspective here: Would Selling the Vatican End World Hunger?" (LINK)
It's kind of like saying the U.S. could end world hunger by selling the Lincoln Memorial, Mount Rushmore, the Capitol and the Smokey Mountains National Park to the Chinese. OK, other than the interesting prospect of having Mao Tse-Tung's visage added to Rushmore, do you think this would do any temporary (let alone lasting) good for the poor and hungry, or the cultural or natural heritage of the world?