TikTok, which has around 36 million users in Russia, said its move was about ensuring the safety of its staff and users.
Since Friday, anyone who writes news deemed false about the military could face up to 15 years in jail.
Among other things, the Kremlin objects to the conflict being called a war, instead calling it a "special military operation".
The BBC and other news outlets have already stopped reporting in Russia, saying they can no longer be independent.
In a series of Tweets, TikTok said: "In light of Russia's new 'fake news' law, we have no choice but to suspend live-streaming and new content to our video service while we review the safety implications of this law.
"Our in-app messaging service will not be affected."
It added: "We will continue to evaluate the evolving circumstances in Russia to determine when we might fully resume our services with safety as our top priority."
Chinese-owned TikTok, which has one billion users worldwide, has been criticised for not speaking out against Russia invading Ukraine, unlike its peers Meta, which owns Facebook and Instagram, and Twitter.
But in a longer statement on its website on Sunday, it described the war in Ukraine as "devastating", adding that it had "brought pain to our community and our people".
Source: BBC
