Imagine a baseball player without his cleats, wearing flip flops to practice. Or a ballerina without her toe shoes trying to dance en pointe in sneakers. Most often, actors need a pair of basic Oxfords (preferably with a leather sole) and actresses need basic "character shoes." All actors should have these every day for rehearsal. The movement one creates in these shoes would be significantly different in another kind of footwear. And our rehearsal room is equipped with the special kind of flooring material commonly used in dance studios.
There is no need to purchase new expensive shoes. I tell kids all the time that the thrift store is an actor's best friend. With the right kind of sanitizer spray, any old pair of broken-in, black Oxfords will work just fine and may even be good-looking enough to use on stage. Theatre people adhere to the "30-Foot Rule" where if something looks good from the audience's perspective, then it is perfect!
Any shoes that are solely for rehearsal purposes may be left in the Black Box after rehearsal so that the actor doesn't have to carry them around all day. However, if they are used for church on Sundays or some other dress shoe event, please provide an easy method for them to be transported successfully back and forth from rehearsal.