If you don't have the time or desire to sell your used furniture, you'll likely want to donate it. Before you look for a charity to take it, you have to decide if you can move it there, or if you need someone to come pick it up. After that, you'll need to do some research in your specific area.
Some charities only accept certain types of furnishings. Many nonprofits also reserve the right to refuse pieces if they look too worn out or damaged. That said, some can be extremely selective depending on their end-use. A charity that aims to sell them will be pickier than one that helps the formerly homeless or those escaping abusive situations.
Finally, if you want to give your furniture away, there's always Freecycle, where you post what you have and someone will come take it.