Dear Landlord won't remove items from previous tenant?
Unfortunately, you cannot just dump or sell the left behind large items and furniture, because you have knowledge that the unwanted personal property belongs either to your landlord or your landlord's former tenant.
The landlord should have moved the former tenant's property to storage, unless the former tenant's lease specifically allowed for the landlord to be free of liability in disposing of personal property abandoned by the tenant when the tenant moved. Most standard lease forms provide for this immunity with language that expresses the tenant's obligation at the end of the lease to move out from the premises, taking all possessions and turning over the house in "broom clean" condition, and deeming any possessions left behind as abandoned property.
More likely, the landlord told you a story, because the "abandoned" property became the property of the landlord. Your landlord made you an "involuntary" warehouseman, at your expense (the cost of the room which you cannot use, and where the landlord stores his property for free.)
To protect your interests and to set up a claim of damages against the landlord, for breach of lease, actual partial eviction and constructive eviction, you need to make a record of your reasonable efforts to convince the landlord to remove his property from your home.
Photograph the storage room with all the large items and furniture in place. Make copies of the photos. Prepare a letter addressed to the landlord. Your letter should remind the landlord that six months passed since the landlord promised to remove his property from the leased house, that the property remains unmoved, that as a result, you lost the beneficial use of one room in your house (if you calculate the lost space against the whole space, you might figure out how much money the landlord owes you for storing his property in your house.) Prepare a few duplicate signed letters with copies of photographs in each. Address envelopes to the landlord, and insert a signed letter and copies of photographs, into two envelopes. Do the same for two more letters you address to yourself.
At the USPS, mail two envelopes to the landlord one via certified mail with a return receipt request and one by regular first class mail with a certificate of mailing. Mail the two envelopes addressed to you in the same manner. Keep all your receipts and copies of the signed letters and photographs. You may be able to gain your landlord's attention.
If you get no relief, you will need to engage an attorney. Your landlord may have committed the wrongful acts of a breach of the lease (he kept for his stuff one room in the house you rented), a partial actual eviction (the landlord actually evicted you from a part of the house) and a constructive eviction (by keeping you out of a significant portion of the leased premises, your landlord constructively evicted you.
Courts in New York differ as to the measure of your damages (based upon the significance or lack thereof of the lost space wrongly taken by the landlord), but your damages could range from gaining a money judgment for all the rent you already paid (your lawyer will likely suggest that you stop paying the rent until your claims are in court), to recovery of a portion of the rent paid. A breach of lease, may entitle you to recover all the rent that you previously paid.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.