I'm developing this Facebook Application and I was wondering if it's possible (and how) to programmatically, through the Facebook PHP Graph API, press some 'Like' button on some page?
Of course, this is optional on my application... I'm still not ready to really explain what application I'm doing, but it would be interesting to code such a feature.
Is it possible somehow?
By your description it sounds like you're trying to get a user to like something without the users knowingly clicking a like-button. This sort of interaction is not condoned by Facebook, I think. There are various black-hatty ways to accomplish this though, one fairly elaborate one is descriped here: http://www.liquidrhymes.com/2010/08/25/smoking-hot-bartender-is-some-smoking-hot-facebook-spam/
UPDATE Sorry, I might be wrong. If you get stream_publish extended permissions from the user, you might be able to like posts on their behalf by doing a POST to /POST_ID/likes. See Publishing to Facebook in http://developers.facebook.com/docs/api
You cannot do this. Facebok wont let you do a POST to /POST_ID/likes, you can only do a get request to retrieve their likes. What you are trying to do is a violation of facebook's TOS. I would suggest just adding a like button and "forcing" them to like before they continue with your application. However, in my opinion even that is kind of silly because they can instantly go unlike it after they have used your application.