I've seen a game for free on iOS when the user 'likes' a page. Though it appears the promotion was run by Apple, not the developers of the game being given away.
How can I make my game free for a Facebook 'like'?
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This was done via a special deal with Apple, and cannot be done in general.
One way you might handle the situation is to make the game free on the store, but have most of the gameplay behind an in-app purchase paywall (such that your app essentially serves as a free demo that allows you to buy the upgrade to the full game). Then, you'd correlate all the FB likes with the Apple IDs of the corresponding FB users and issue the tokens for the upgrade IAP to those Apple IDs.
Unfortunately, there is no mechanism to do that Facebook-to-Apple-ID join, so you'd need to do it manually or build your own webservice that users used which would collect their information, do the "like" on their behalf, and then store the correlated IDs in a database you could process.
But even then, there isn't a way for you to issue tokens for IAPs (or apps themselves) arbitrarily. You can request promotional codes from Apple which could you then distribute to FB likers, but the number of promotional codes you can acquire is pretty limited, and unless you expect your total userbase to be well under that limit (less what you'd want to use for actual promotional purposes), doing this would probably be a terrible idea.