The video-sharing giant said it wanted to ensure the safety of staff and users as it assessed the new law.
Since Friday, anyone who writes news deemed to be fake about Russia's armed forces can face a lengthy jail term.
The BBC and other news outlets have stopped reporting in Russia, saying they can no longer be independent.
There are about 36 million monthly users of TikTok in Russia, but developers there recently launched a rival service.
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.
The site has also been accused of enabling disinformation during the conflict, as videos from both Russia and Ukraine spread rapidly.
TikTok says it "doesn't break down user numbers by country" but we know that Russia is one of the company's top territories.
It's thought that around 36m people use the app and TikTok's new rules reduce them to spectators of other people's content.
TikTok says it's making the move to protect it's users from breaking Russia's new "fake news" laws.
But it will no doubt help the company too.
Moderating social networks during times of crisis is hard enough, but having to vet content under the new draconian laws in Russia puts all platforms in a tough position.
I wouldn't be surprised if other apps took similar measures to protect themselves.
Russia has many high profile creators who make their living from TikTok so we can expect even more people to be motivated to protest themselves against the "fake news" laws.
The Kremlin may also lose out in the information war too with less pro-Russia content on people's timelines.