Can our landlord make us move out of a handicapped apartment if a handicapped person needs to move in?

You have a lease to rent that unit. You are not legally required to move unless you do not have a lease - then he needs to give you one month notice. If your landlord wants you to move before the end of the lease, tell him he has to pay your moving charges and utility transfer charges and any other cost that may result from having to move to a different unit. Maybe he can offer you a larger unit for your troubles.