No, Facebook apps are something completely different, they are embedded directly into Facebook pages. Facebook dictates here what an app can do and what it cannot. In particular, a Facebook app indeed cannot run JavaScript because that JavaScript would execute in the context of facebook.com. Consequently, using Adsense is impossible.
Your application however is a regular web application that is available under your URL, not Facebook's. Facebook login or a Facebook "like" button don't change that, you can still run any JavaScript code meaning that you can use Adsense.