Although it hasn't quite made it to national holiday status, Residents’ Day, established more than 60 years ago in an attempt to strengthen Tokyo's communities, sees many museums, galleries and parks (generally those run by the Tokyo Metropolitan Government) offer free entry. It takes place annually on October 1, and most schools are closed on this day, so expect crowds. (Full list below.)
Hamarikyu Detached Garden, Kyu Shiba Rikyu Garden, Koishikawa Korakuen, Rikugien, Kyu Iwasaki-tei House & Gardens, Mukojima Hyakkaen, Kiyosumi Garden, Kyu Furukawa Gardens, Tonogayato Teien, Jindai Botanical Garden, Tama Zoo, Ueno Zoo, Tokyo Sea Life Park, Inokashira Nature & Culture Park, Yumenoshima Tropical Greenhouse Dome, Tokyo Port Wild Bird Park, Edo-Tokyo Open-Air Architectural Museum, Tokyo Photographic Art Museum (permanent collection and two special exhibits), Tokyo Metropolitan Art Museum (Hiroshi Sugito exhibition only).
Note: Museum of Contemporary Art Tokyo, Tokyo Metropolitan Teien Art Museum and the Edo-Tokyo Museum are closed due to renovations.