Ultimately the most important key to running experiments is to install the Optimizely snippet on the site on which you want experiments to run. Depending on how the page is created and where the source code lives, that will dictate whether you can install the optimizely snippet and execute experiments.
The standard in-Facebook method of creating pages doesn't give you access to source code and the page lives on the facebook.com domain, so this method would not allow you to A/B test. I know there are other methods to developing and creating Facebook pages and apps, but my direct experience with them is a little rusty, so I'd love to hear other community members' ideas.
A possible workaround would be to create two disitnct Facebook pages, and then use Optimizely on your regular site to perform a redirect test to each one. Or, run a simple A/B test where you change the link behind a CTA to each one of your pages.
Solutions Architect | Optimizely, Inc.