If I may add one of my experiences with Twitter (it was a year and a half ago):
I had a guest blogger with many followers on Twitter (and account which is visible on several news related sites) who twitted his particular post and this brought me several hundred visitors to this post with few comments, likes, shares etc. too.
This tweet was also retweeted about dozen times.
Next day my blog jumped from around 20th to 5th position for my main keyword (slikanice - picture books) in google, although this word was probably not even mentioned in the guest post. In next days the position slowly dropped back and after two or three weeks reached number 20 again.
This is only isolated event, but I believe a tweet with such response can help in rankings. Of course this will also have only temporary results (if something dramatic doesn't happen with visitors when they land on the tweeted address, high CTR for instance).