I don't know how to fetch just all people that have pressed the like button for an object, but you can subscribe to the like-button-press event. You can then use the information about the current user, and save it to your own database.
The event that you are listening to is edge.create, and you do that with FB's own FB.Event.subscribe.
It could look something like this:
window.fbAsyncInit = function() {
FB.init({
appId : 'xxxxxxxxxxxxxxxxxxxxxxxxx',
status : true, // check login status
cookie : true, // enable cookies to allow the server to access the session
xfbml : true // parse XFBML
});
FB.Event.subscribe('edge.create', function(href, widget) {
FB.api('/me', function(response) {
alert(response.first_name + " " + response.last_name + ":" + response.id +" pressed " + href); // Replace this with an ajax call to your server
});
});
};