I am writing a book which tells the story of a year in a particular niche. I want to include some tweets in the book as images like you see embedded in web pages all the time. Do I need to ask for permission to do so, is there any kind of copyright issue? They will be part of the book and appear in the context of the timeline I am working through in the narrative.
I would suggest using a non-descriptive social site, which may work as stand in for the top social networks as needed.
Anyway, on your question...
On using real tweets
The author of the messages is the author of the messages and has all the rights any other author would have (unless they took them from somewhere else, in which case they are not the authors).
Then how can Twitter publish and distribute the tweets? When you sign up for a Twitter account you accept their terms of service, in which you grant Twitter a license that allows them to do that. However, by doing this, people license their messages to Twitter, not to you. Review Twitter Terms of Service (look under numeral 3. "Contents of the Service", title "Your Rights and Grant of Rights in the Content").
This means that - unless the author has said otherwise - you do not have a license to what they tweet.
In addition, of course, Twitter presents the tweets to you under copyright. Therefore, yes, the tweets are under copyright and you should ask the author for permission to use their work.
You can always have made up messages.
On using Twitter user interface or logos
Just as anything else on the web page, the logos and other interface elements are under copyright (also mentioned in the linked Terms of Service).
Then, how is it possible to embed tweets in other pages or use the Twitter logo? That is allowed as long as it follows Twitter's Developer Agreement. Which I will not summarize here, I will just tell you to pay attention to what it says about "Twitter Marks".
You can contact firstname.lastname@example.org for inquiries.