Publish tweets in drupal 7 site in real time
Try Selective Tweets. As written in comparison to another Twitter module (Tweed Feed), Selective Tweets renders Tweets almost realtime:

Tweet Feed does periodic imports of Tweets and turns them into nodes,
while Selective Tweets renders Tweets almost realtime and saves none
of them in the database as content. This is also the main difference,
with the effect that Tweet Feed has a significant delay on the
realtime Tweets. On the other hand, the Tweet Feed created nodes can
be displayed in Views which gives big flexibility in the way of
displaying Tweets.
Use Selective Tweets when you want a light weight
manner of displaying Tweets without database storing and you want no
major delay in displaying realtime tweets.