Art Museums
European art museums would be empty because all of the art would either be in the Führermuseum or burned. The Louvre, Museum Island, and Jeu de Paume would be empty and would "not exist". All of the art that Hitler had taken would only be able to be seen at his Führermuseum. Only Germans would be able to see the glorious art and everything done with the art would be his decision.
European History
If the art was burned or destroyed; a huge chunk of European art and history would be gone forever. Many of the main tourist attractions in Europe are museums. Museums are one of the places that people go to learn about their ancestry and past history. Most of the paintings that were made before WWII were portraits of people that the artist thought would be a good artistic influence, if these paintings were the destroyed the documentation of these famous people would have never been made.
Cultural Heritage
Countries want the art created in their country and home countries of its creator, there has been many disputes after wars of where the art belongs usually resulting in a fight. Some of the art pieces have been in one location for hundreds of years, in the disputes they are moved to other locations thus disturbing them potentially wrecking them. Keeping these priceless painting is one location is best for painting to keep it in good condition and the people so there is less fighting.