The central government has pledged almost $100m in aid so far.
Kerala Chief Minister Pinarayi Vijaran said: "I wish to inform all that the state government will now take very strict action against such people."
The Indian Army in turn exposed a fake video where a man in military uniform is seen talking about the relief operations and criticising the Kerala chief minister.
"Imposter wearing army combat uniform in video spreading disinformation about rescue and relief efforts," the army tweeted.
"Every effort (being made) by all and Indian army to overcome this terrifying human tragedy," it said, urging people to inform authorities about suspicious news.
The deluge of false news, videos and claims also comes as the Indian government steps up pressure on Facebook, WhatsApp and other platforms to act against fake news, blamed for inciting more than 20 lynchings in the past two months.
Hoax messages about child kidnappers being seen in villages spread rapidly on platforms such as WhatsApp, which boasts more than 200 million users in India.
Mobs quickly gather to target the alleged intruders. More than 20 innocent people have been killed across India since June.
Facebook is investigating intensively to counter fake news in India and other countries, hiring teams to track content and linking with fact-checking groups, its officials say.
The Indian government demanded that Facebook-owned WhatsApp clamps down on fake news in talks this week with its chief executive Chris Daniels.
Information Technology Minister Ravi Shankar Prasad said he was assured that WhatsApp "will soon take steps" to comply with the demands.