For an entity to be classed as alive, they need to respire in some form. For homo sapiens, this involves breathing. From a musical aspect, though, it could be argued that breathing in recordings serves a different purpose to that of providing life.
For such an incoherent process, it can be somewhat facile to overlook when working on a musical project. On one hand, allowing a vocalist's breathing to come through during editing can aid the composition in making it sound less artificial. On the other hand, there may be situations in which the vocalist's breathing detracts from the overall performance.
A possible example of breaths detracting from a performance can be found throughout the composition by Muse "A Map of Your Head". In opposition of this, though, the listeners may argue that this is part of Muse's sonic aesthetic, and that anything other than Matthew Bellamy's loud breathing "...wouldn't sound right.", as some sources have described.
In contrast to this, having no breaths or almost inaudible breathing in a vocal performance can add a sense of artificiality to a performance; despite the negative connotation, this may add to the sonic characteristics of a song or particular genre. In the Pop Punk song "Knuckles" by Moose Blood, the breathing that you would expect to hear between each line in a verse doesn't appear to be made a feature within the song; whether it was made so during the editing stage, or not entirely captured during recording is an ambiguous question.
Arguably a happy medium between both sides, is Jeff Buckley's rendition of "Hallelujah". Aside from the initial breath at the beginning of the song, each breath sits naturally between each phrase without making the Jeff's breathing the focus of the song, but simultaneously, it adds an organic transition between phrases.