I don't believe the author has any legal right to payment for someone copying their tweet (their tweet log is a different story), as you said quote compilations

I don't believe the author has any legal right to payment for someone copying their tweet (their tweet log is a different story), as you said quote compilations get away with it all the time. However, copyright still stands and the author of the tweet has the legal right to be attributed alongside the quoted tweet, but as the 'work' would unlikely be sold for profit then the author wouldn't get a penny from a judge anyway. The difference between a tweet and Hemmingway's 6-word novel is that he could (and did IIRC) make money off of it.