“Traditional” restaurants, some museums and galleries, most Japanese B&Bs (“ryokans”) have floors covered in tatami mats. Those mats are made from very soft, woven straw. Shoes or bare feet can easily damage them. So you will be asked to take off your shoes before entering. Some places may have lockers in which you can store your shoes but your shoes are almost always perfectly safe just being left where you take them off.
HINT – to avoid embarrassment, wear good socks without holes
HINT – consider wearing shoes that easily slip on and off
HINT – consider elastic laces which enable any shoes to be easily slipped on and off without undoing the laces every time
Japan Tourist Tips 