Should the particle 的 be omitted among nouns that refer to other nouns, such as phrases like "the plot of the film", "the head of a giraffe", etc.?
If you can rephrase it in the genitive case, no. The film’s plot, the giraffe’s head, etc. Then use 的.
On the other hand, if you have a more complicated sentence, you can and should omit all but the last 的.
The plot of the film of the giraffes: 关于长颈鹿（的）电影的情节。
The particle 的 ties two nouns closely. For instance, 電影的情節 (the plot of the film) emphasizes the plot of a film but that 電影情節 is about the plot and the film which is loosely connected and the meaning of each noun is stressed.