First, please run our the Error Check tool in our Autotune backend area of JFBConnect. Second, please check your error console for any Javascript errors that may be occurring on the site, which may interfere with both Login and Like.
If looking at these two things don't indicate any errors, could you provide the exact URL's where you are seeing these errors? As every site is configured differently with many templates and extensions, it would be hard to pinpoint the problem without more specific details.
-Melissa