Is it OK with facebook policies to make a "like" button that will disappear for user when he use it?
If not - is there some other button that can be placed in other site (not on facebook) that will act like a "like" button but that cannot be "unliked"?
I think it is fine if a user can not unlike something. This would be similar to online voting - like the survey questions on news sites that ask readers for their opinion. Once the user picks their response, their vote is cast and they can't undo it. Just think of the like button as the option to vote for "I like this" or "I don't like this". This would be acceptable if they are likely to only see what they are liking once and probably won't return again, like a news story. When they first see/read it, they can decide if they like it or not, and they probably will not go hunting to find it again to unlike it.
However, my response does not apply if this would control some kind of user preference that they may want the ability to change in the future. For example, I can give a song on Pandora a "thumbs up", so it will be played again in the future (as well as others like it). If the next time I hear the song, I don't like it anymore, I can click the "thumbs down" button instead so I won't hear it again. Although, this is a bad example to illustrate my point, since you actually can't unclick the Pandora buttons.
Either way, I would not recommend making the button disappear (unless you replaced it with some text like "You like this"). This would be confusing to the user if it was suddenly missing, and they need the visual confirmation of their selection, even if they can't change it. Instead, I would recommend disabling the button on click.