I think it really varies depending on school. Some wait-lists are ranked, so it won't matter whether or not you submit a LOI since they have to follow the ranking. For unranked lists, I feel like it would be helpful, especially towards the end of the cycle after you've interviewed at other places because it would allow you to write more about what specifically makes their school unique. If they know you've been accepted at another school but would rather attend their school, they might be more likely to pull you off, especially if the school you were accepted to is higher ranked.
