The limited-time deal is only good for a small 12-ounce hot coffee and only during breakfast hours (which tends to run until 10:30 or 11:00 at most BK restaurants). A small cup of coffee typically goes for $1 at Burger King.
The chain switched to new, Burger King-exclusive Smooth Roast coffee by Seattle's Best Coffee last year and is looking to continue promoting it as the new year begins. Smooth Roast is described on the Burger King website as "a 100% Latin American Arabica blend characterized by sweet, nutty and chocolaty notes that delivers a balanced mild flavor."
I reviewed it last year and thought it was decent enough.
The free coffee offer runs through January 29, 2014.