I have visited hundreds of museums in dozens of countries, but I don’t recall ever being required to wear special footwear. However, I recently stumbled on this old New York Times story about the practice. From the New York Times: “Some museums don’t want you clomping around in your street shoes, damaging their famous floors. So they offer specialized footwear.” Have you ever worn museum slippers ?
Travel Between The Pages 
Yes. Parts of St. Mary’s Church, Studley Royal near Ripon (https://www.english-heritage.org.uk/visit/places/st-marys-church-studley-royal/) – a William Burges masterpiece no longer in use for regular worship.
Very common in historic house museums. Instead of having roped off areas to keep people from damaging rugs and floors, museums moved to letting people experience the whole room and wearing protective covers. A great improvement over just peeking in through a doorway!