If you've logged into your Weebly account through Facebook and decide you no longer want to do so, you can disable this in a few short steps.
First, go to your Account page from your Weebly dashboard.
Click the Edit Profile link, then on "Connected" next to Facebook - you'll be prompted to Disconnect in a pop up, so go ahead and click that. Don't forget to save in the Edit Profile window when you're done making changes.
Head over to your Facebook account next, and click the arrow in the upper right to access the Settings page.
From there, click on Apps in the left sidebar and locate Weebly in the list. Click the X to the right of the icon to completely remove Weebly, or click on the pencil to make changes to the permissions.
You should be all set after that, and if you change your mind you can always set up Facebook login again by clicking the Facebook login button at the sign in screen.
Sources: Weebly Help Center