I am a landlord in New Jersey, I rented my apartment for a year then it turned into a month to month lease, the tenant told me about three months ago that he would initially be leaving in June but since he was looking to buy a house and it took longer than expected it turns out he will be leaving in August, there was only verbal communication as to the move out dates and we both agreed we would be flexible with each other. My question is this. His rent is due the 24th of each month, he has paid the full months rent to August 24th but is moving out on August 15th am I obligated to give him back the remaining days from the 15th to the 24th?
Thank you for your time

Without a written notice from him you're not OBLIGATED. At this point you've both dropped the ball by not communicating in writing.

How I would handle it... If I felt like he gave me a valid, notice by July 15th that he was going to be out by Aug. 15th and he left the place like he found it I'd probably work with him. If he's been going back and forth on dates to the point where I couldn't count on anything he told me, etc. I probably wouldn't refund anthing.

Get everything in writing from now on.
Is he asking for the 16th - 24th back?
Also, what does the lease say regarding notices?

Like Rockin, I would say it depends on when he gave you the 8/15 date. If he gave you enough notice, then I would give it to him, if he didn't, then no.

You are not a hotel or a motel. You are a landlord renting on a month to month basis. If you want to give him money back, then pro-rate the rent. You are not obligated to do so. He has a verbal contract to rent until the 24th of each month. However, you can not rent the same property to two different families at the same time. Therefore, if you rent the unit and they move in before the 24th of the month, you would owe the former tenant for the days it was occupied by another family.