The title says it all. If you add a FB like box on a website to gain more followers, is it a requirement that you're logged into Facebook in order for it to show on your website?
No you don't. Logic dictates that they can't require the site owner's account to always be open. Unbanned certainly, but not open/logged in.
I mean for the visitor to be logged into FB. When I am not logged in, it doesn't show the like box on my site. I did, using this page as a guide https://developers.facebook.com/docs/plugins/page-plugin
Yep, after all if the visitor isn't logged in how will Facebook know to give credit for the like, and whose newsfeed will it show the like in?
Yes the user must be logged in, or more accurately the user MUST NOT be logged out of facebook-- meaning the user doesn't have to have FB open in another tab/window for their browser session to stay active as long as they have NOT intentionally logged out of FB.