The scene in which Faramir is leading a battle and Denethor is eating the tomato while Pippin sings. What is the significance of that? Ostensibly it's to show that Denethor isn't interested or has already given up on his son, Faramir, but why was it shown that way in particular, with Pippin singing and the extreme close-up of eating?
It shows how deranged Denethor is
Peter Jackson: I remember we came up with the idea of the eating during this, too, because there's something very nasty about eating and violence, you know? And a sense of violence, a sense of killing his own son while shoving strawberries and tomatoes into his mouth and it's just, I don't know, it's something that—
Fran Walsh: It's cracking, and spitting, and—
Philippa Boyens: Yeah, the tomatoes, ugh.
Fran Walsh: It's very venal.
Peter Jackson: Yeah.
Philippa Boyens: It is.
Peter Jackson: It's a bit uncomfortable— I think, you know, it makes the audience more uncomfortable, the fact that he's stuffing things into his mouth, than it would be if he was just sitting in the chair doing exactly the same dialogue. It's sort of the fact that he is sort of enjoying— because usually in situations like this, when there's life and death situations and it's war and it's, you know, a huge threat, you would lose your appetite, and the fact that he's sitting there kind of eating, even that helps sort of show how deranged he is and how sort of disconnected from it all he is.
The Return of the King Extended Edition - Director's Commentary [1:32:31-1:33:32]
The Red Tomato Pulp Represents Blood
The movie was rated PG-13 in the United States for its battle sequences, preventing them from showing as much blood and gore as some of Peter Jackson’s earlier movies. This scene made striking use of visual symbolism, music and lyrics to represent the decadence of both Denethor’s lifestyle and the lives he was wasting.
Peter Jackson significantly altered the character from the books, in order to add this scene. In the books, Denethor II was described as “a man of another sort, proud and subtle,” whose lifestyle is almost ascetic (his chair is “black and unadorned” and his chambers “sparsely furnished”). Neither is there any scene in the books where Denethor asks a Hobbit to sing for him. This was added by screenwriter Phillipa Boyens, after hearing Billy Boyd sing at a karaoke bar.
It is worth noting that Peter Jackson made the decision to adapt Tolkien’s “A Walking-Song” from chapter 3 of Fellowship of the Ring, by removing all lines about making it back “to home and bed,” and altering the line “Away shall fade, away shall fade,” from the middle of the song, into the closing line, “All shall fade, all shall fade.” Boyd, who was asked to write his own music, wrote several different Celtic-style melodies for the song, and they chose one as different from the festive drinking-song before as they could get. This changed the song into a melancholy dirge, fitting the intended meaning of the scene.
Edit: This answer has turned out to be controversial, because it doesn’t cite any source for this interpretation. Honestly, that was because I thought the scene speaks for itself. A great many other people have observed that the tomato pulp running down Denethor’s chin, as his callousness sends his men riding to their deaths and Pippin sings of death, looks just like blood. There’s no hidden reference that needs explaining. You can see it with your own eyes. It looks that way to you, or it doesn’t.