Liking a page on behalf of a user?
You can't like a Page on behalf of a user (Bugzilla discussion). You can, however, like posts, comments, and photos on behalf of a user.
Edit 7/9/2012
Since bugzilla no long exists, the bug linked above is inaccessible. Google doesn't have a cached version of the page, so I ran another search. The best thing I could come up with was this Google Code Discussion regarding the ActionScript API.
Facebook makes brief mention of Publishing likes via the Graph API in the documentation, but doesn't say one way or another whether you can like a Page on behalf of a user - just "Objects" which (probably arguably) are not "objects" in Facebook-lingo.
My thought is, the API to like page is available, but is only offered to white listed applications (such as, the Facebook iOS and Android applications) written by "special" publishers. There's obvious reasons why Facebook wouldn't want/allow developers to create like connections on the graph. It would be taken advantage of by spammers and other nefarious developers and would deteriorate the meaning of what a "like" represents for a page on Facebook.
My guess is, you'd have to make a pretty strong case to Facebook about why you need/want access to the Page's Like connection (for publishing) before they'd even consider giving you access. I'd also guess that they'd want to verify that you're doing only user initiated like creations (in such a way that the iOS application would handle it) so as to protect the reptutation/meaning of a "like" action.
Actually this is NOT true, but you have to do a complicated Javascript / UIWebView process in order to display a Facebook 'page' of JUST the like button on your view, and this like button you can configure in the JavaScript / Objective-C (using string replacement) to be any Facebook page url you like.
Facebook's platform policies don't allow for a web-based like button aside from using the officially supported options
Those options doesn't require using OAuth or the Open Graph api. However, facebook just added support for mobile apps to send like actions through opengraph.
I'm not sure if they intend to allow sites to customize their like buttons or just apps...