In the log there are errors concerning multiple definitions of generic collection classes. I notice that another post in the forum has a similar issue that was fixed in a future Beta.
Could be the same issue. Is the Facebook C# SDK using generic collections? FB.cs?
Regardless of the compilation errors (which are bugs and should be reported, please), the current Facebook SDK is implemented in platform specific ways which would not work on WebGL. However, Facebook is AFAIK working on a WebGL implementation, so stay tuned for a version which will work.