The only time I would talk about obligation is if you had entered into some agreement with the author to review or give feedback. I don't think purchasing a book or taking advantage of an Amazon freebie constitutes an agreement to review.
That's my take anyway.
As for author's responding to tweets asking to spread the word. I don't think there's anything wrong with that per se, but as a reader you shouldn't feel any sense of obligation. The author is just hoping to use any means possible to spread the word as they know full well how important word of mouth is - especially the indie authors. There's a reasonable risk to them if they go heavy handed or lay guilt trips. Instead of possibly getting the advantage of some good word of mouth, they could alienate the only fan they have.