The Met recently made all of their public-domain works available through their beautiful website. It's really remarkable — you should take a look for yourself. All of the searchable works are available as high-res downloads. The Creative Commons Zero licensing permits using, sharing and remixing without restriction.
They' even published data to GitHub that, while not quite a full-blown API, is enough a stepping-stone that I'm sure people will start building interesting things from it in no time.