The fine arts constantly struggle for funding, perhaps due to their general inability to compete with modern sources of entertainment.
In art museums, commercial sponsorship could take the form of (non-destructive) modification to the works of art themselves. For example, the Mona Lisa could be holding an iPhone (an idea which has been done before: https://www.google.com/search?tbm=isch&q=mona+lisa+iphone), or one could spot a Radio Shack in the nightmarish hellscape of Hieronymus Bosch’s Hell (http://upload.wikimedia.org/wikipedia/commons/f/f6/Mad_meg.jpg).
For flat artwork, sponsorship images could easily be added by using a glass overlay with the desired promotional material painted on. See below for details:
Fig 1: A clear overlay (perhaps a piece of glass, or an animation cel) would be slid over the piece of artwork in question. In this example, “The Scream” is modified to be chomping on a delicious hamburger. Perhaps this particular overlay would be a McDonalds ad, which might encourage Burger King to buy a competing overlay for another famous painting at the same museum.
Fig 2: Side view of the above image: A is the clear overlay, B is the painting.
This is a great idea and you (assuming you are a museum director or curator) should apply it right away!
PROS: Saves fine art from destruction, brings more visitors to art museums.
CONS: Could make regular non-sponsored museums seem boring in comparison.