Dragons can't eat tower pieces or barns (only followers). Dragons can eat followers on the tops of towers. This is discussed in the huge annotated rules clarification on Board Game Geek, which has the following to say.
If a follower is deployed to the top of a tower, the construction of the tower is finished
and it may not be built any further. The follower remains on top of the tower until the
end of the game and is not removed or returned to the player unless eaten by the
dragon or captured by another tower. This action may be useful when a player
wishes to protect a more valuable follower from capture.
A barn cannot be removed by the dragon (from The Princess and the Dragon), nor taken prisoner by a tower (from The Tower).
Some relevant related rulings (footnotes in the document):
(105) Q: If there are 2 followers on 1
tile (tower and city) does the dragon
eat them both?
A: [It eats them ] all.
(116) Q: If the dragon can eat a
follower on a tower, can the follower
also be seduced by a princess?
A: The tower is not a part of the city: they are two separate areas.
(133) Q: If you place a dragon tile and add a piece to a tower, what happens
first: follower capture by the tower, or dragon movement?
A: Prisoners are taken first; placing a tower piece is an
alternative to deploying a follower, so occurs before dragon movement.
Dragon food clarifications can be found on p.68-70. To summarise:
- Dragon food: Follower, Big Follower,
Mayor, Wagon, Builder Pig, [Follower on] Tower
- Not dragon food: Barn, Count,