I live in NYC and am moving out of my apt on May 23rd but my lease extends (and I am fully paid) through May 31st. I gave my landlord proper notice and he has already rented my apt (and actually asked for me to move on the 25th once he found the new tenant). NYC apts turnover fast, but I expect a day or two of cleaning time. Are there any legal grounds for me to ask him to refund the last week of rental expense -- particularly for the days where he is receiving double rent?

Based on your post, you have a legal lease extending to May 31. I would inform the landlord that although you will be effecting the major part of your move earlier, you will not be vacating until that date. If he wants, you can vacate a few days early in exchange for a refund equal to the daily rate of the rent. If he refuses, you have no legal grounds for a refund, but he has no legal grounds to take possession of the apartment until May 31.