Change Windows Server 2008 R2 Full to Server Core

This is not possible. You will need to reinstall the OS.

Technically it is possible, but Microsoft never bothered to spend the money or manhours to implement such a niche feature. I, for one, am glad.

Plus, for a Server install, I wouldn't trust the process even if they did implement it. There are just way too many test cases to consider and test, too many for a company even like Microsoft to cover.