First of all, decide which furniture you want to donate. After making a complete list of the items, make sure the stuff is not too damaged. It would not take much time, if you repair small damages to give the deprived one the real joy. Also clean it up properly.
In your area, there may be a few Goodwill stores. These stores usually have an arrangement with the local shelters. The Goodwill stores receive all the stuff from you, and subsequently pay the shelters in the form of cash or other stuff.
If the Goodwill route is not available, you can contact the shelters directly. Usually, the workers in the shelters pick up the furniture from your place, if they have the resources. You may find plenty of charity organisations on the web. Select the one that you feel deserves the most.
If no plan works out, you can simply donate it to a church. A lot of churches do accept such donations. Apart from furniture, they accept appliances and other household items. Every organisation will gladly acknowledge your offer.