If the previous tenant does not move their things out by moving day, they are most likely breaching a contract.  If this is the case you should contact your landlord and ask him to take care of it, as he is responsible for upholding and closing any contract with previous tenants.  The landlord will most likely require authorization to move any abandoned possessions.  You can also send a certified letter to the previous tenant stating that their possessions will be disposed of if they are not picked up by a certain date.