We had the idea of running Twitter accounts that tweet successful requests from the RSS feeds provided by Alaveteli.
TL;DR: No.
Based on our current metrics, it doesn't seem like links posted through a Twitter bot account for any significant traffic.
The links tweeted are the Request events (rather than a direct link to the request) so we can look at URLs starting /request_event/. (Note the image below uses a Regexp, but that didn't seem to return any results for NZ).
FYI NZ already do this using fyi-squeaker, so would be a good indicator of whether its worth pursing elsewhere. They have a substantial follower count (nearly 700 in April 2016) so its likely people are at least seeing the tweets.
Between January 2009 and February 2016 only 0.01% of site traffic came from /request_event/ URLs. For 2015 12% came from social media in general. Of that around 70% from Twitter (so 8% of the total).
We've also set up @Ki_Mit_Tud_Bot to try the same in Hungary. Between March 2016 and 26 May 2016 0% of traffic came from /request_event/ URLs, therefore no traffic has come from Twitterbot.