No. Facebook does not charge publishers for a share/post to Facebook button. Here is the official link: Share Button. On clicking on "Get code", you will get a code snippet that you can include for free in your website. If a user decides to use a "Share/like" button, that is a win-win for both Facebook and the publisher, and it does not make any business sense for Facebook to charge for this feature.
In fact, on Allrecipes, if you wait for the full page to load (The page load time is definitely more than usual.) , you will see the FB icon appearing in the list of social plugins.
See question on Quora