Usually not in current academic style. MLA style uses italics, not underlining, for the titles of longer works, and a newspaper counts as a longer work, so you would write The New York Times or Los Angeles Times in italics. APA and Chicago styles also italicize newspaper names.
Underlining was the older substitute for italics when papers were handwritten or typed on typewriters that could not produce italic type. Today underlining is generally reserved for handwritten work or for cases where an instructor specifically asks for it, and an article title inside a newspaper goes in quotation marks rather than italics.
Source: Purdue OWL: MLA General Format
Image: A stack of newspapers, Daniel R. Blume, CC BY-SA 2.0, via Wikimedia Commons.
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