If you’re writing for fun, a blog, or a school essay: italicize newspaper names. It’s safe, it’s stylish, and it makes your text look classy. If you’re writing for a specific publication, check their style guide. But honestly, unless your boss is a grammar guru, nobody will scream at you for italics.
And if someone does yell at you? Just say you were following Chicago style. Then take a sip of your coffee and walk away. You’ve earned it.
Now go forth and italicize like a champion. Your sentences will thank you.