I recently 301 redirected an old domain to a new domain...
Facebook:
Unfortunately, Facebook Likes don't pass through. I have read a few articles that recommend editing the meta property og:url to the old URL, as well as changing the data_href on the Like button to the old URL. But I haven't seen it work, and the Facebook Object Debugger throws an error.
Google+
The +1's from the old domain did transfer to my new domain. I can tell that the +1 counter didn't start from 0 again, plus, since I +1'd a couple of my own pages last year on the old domain, I can see those +1's in my Google+ profile. It shows me that I did indeed +1 pages on the old domain last year, and if I go to those pages on the new domain now, the +1 button is red and lets me know that I publicly +1 it. It looks like those transfer just fine.
So Google+ share counts seem to transfer, but not FB Likes. Is there a way that Google can know a page on the old domain has a lot of FB likes, and even though the count doesn't transfer to the new domain, they use the old domain social signal and pass its authority to the new domain? The more I think about it, probably.
My old domain had importance, and soon after the 301 my new domain seemed to reap a lot of the old domains value. Even if Google can't pull up my old FB Like count in the same way that I can't, there's probably still a social signal value that Google has stored.
So the more I think about this, even though my FB Like Count for a specific page went from 2000 Likes on the old URL to 0 Likes on the new URL, I think Google still passed that authority to the new URL. I don't have hard facts, just what I've observed. I do wish I could get my old Like Counts back though.