Close reading is probably the most important legacy of the New Criticism. Historically, it was a reaction against criticism that was not rigorous enough ("impressionistic") but it did not disappear when New Criticism was replaced by structuralism and the deluge of theories that have followed since the 1960s.
Avner Shahar-Kashtan says that it arose from religious traditions without explicitly mentioning either Biblical hermeneutics or hermeneutics in the sense of "the theory and methodology of interpretation". Jonathan Culler uses the term "hermeneutics" refer to a type of criticism of which New Criticism is an example, and contrasts it with "poetics" (italics in the original):
[Poetics is] an understanding of the devices, conventions and strategies of literature, of the means by which literary works create their effect. In opposition to poetics I set hermeneutics, the practice of interpretation, whose goal is to discover or determine the meaning of text.
In principle, these two enterprises are diametrically opposed: poetics starts with attested meaning or effects and seeks to understand what structures or devices make them possible, whereas hermeneutics argues about what the meanings are or should be. Anglo-American New Criticism had made hermeneutics or interpretation the orthodoxy of literary studies, and I argued that rather than aim at discovering new yet sound interpretations of literary works, literary studies should seek to understand how produce the effects they have for readers (effects such as meanings).
(Jonathan Culler: "Preface to the Routledge Classics Edition", Structuralist Poetics, Routledge, 2002; the book was originally published in 1975.)
In the context of literary criticism, both hermeneutics (e.g. New Criticism) and poetics (e.g. structuralism) require, or at the very least benefit from, a close attention to formal characteristics such as style, metaphor, rhythm, rhyme etc in addition to meaning. (This is where New Criticism seemed to have gone beyond the religious traditions from which, according to Avner Shahar-Kashtan, close reading arose. I am not aware that the religious hermeneutic traditions devoted much attention to language as a reality in itself as opposed to a vehicle for something distinct from language as such.)
Culler also discusses close reading in Literary Theory: A Very Short Introduction (Oxford University Press, 1997), more specifically in chapter 3: "Literary and Cultural Studies". Culler cites the introduction to Cultural Studies, edited by Lawrence Grossberg et al (1992), which states: "although there is no prohibition against close textual readings in cultural studies, they are also not required."
This assurance that close reading is not prohibited is scarcely reassuring to the literary critic. Freed from the principle that has long governed literary studies—that the main point of interest is the distinctive complexity of individual works—cultural studies could easily become a kind of non-quantitative sociology, treating works as instances or symptoms of something else rather than of interest in themselves, and succumbing to other temptations.
The implication of this comment is that close reading is (or was) something like the foundation on which literary critics build their analysis (even though the process of close reading is not necessarily visible in the resulting interpretation).
You don't need close reading if you are just reading for pleasure or if you use literature as "a tool to help us live and die with a little more wisdom, goodness and sanity" (What is literature for? by The School of Life). You need it if you want to increase your understanding of how literature works.
You also don't need it if your method of analysis relies on computational methods to explore larger corpora of texts rather than interpreting individual text. Distant reading (no pun intended on my part), for example, is described as "an approach in literary studies that applies computational methods to literary data, usually derived from large digital libraries, for the purposes of literary history and theory" (quoted from Wikipedia).
Conclusion: whether close reading is "necessary" depends on your method of analysis; it is unnecessary if analysis is not what you are interested in.