If a prospective tenant gives the landlord a deposit to hold the rental property for them and they are acceptd as tenants, and after they agree on a move in date and all is done, if the accepted tenant pulls out of the agreement does the landlord have to refund the deposit that was given, when it clearly states in the rental app. That all deposits shall becom non-refundable if an accepted tenant does not take possession by date agrred upon? I really need some answers here if anyone knows the laws for this.

If they paid, and rented, they are liable for the rent on the apartment, can I assume the lease was signed?

But a non refundable depoisit is just that, not refundable

when it clearly states in the rental app. that all deposits shall becom non-refundable if an accepted tenant does not take possession by date agrred upon?

As long as this is not contradictory to the state's landlord / tenant laws in some way, then NO, the deposit to hold the rental would not be refundable if the prospedtive, approved tenant decides not to take the place.

There's a "sticky" at the top of the forum with a link to each state's landlord / tenant laws.