There are a few options here.
phpbb in itself does not support this feature.
However there are some add-ons that would allow this.
And here the problem of choice arises.
There is a certain add-on that will make a connection with fb and will put it that you liked it on fb as well (I think).
But this would be a bad choice, as fb sucks and people without fb might in fact not be able to use it.
The more sensible solution here is a karma mod.
Since in smf forums there was a standard option, phpbb followers created something similar to compete.
In this case everyone, apart from guests and bots could be able to like or not like certain posts.
The latter would be a more suitable solution in the lines to what you want.
http://www.phpbb.com/community/viewtopi ... 0&t=559069