Police complaints have been filed against Mr Bhat by Maharashtra's ruling BJP and also Raj Thackeray's Maharashtra Navnirman Sena (MNS).
The video, captioned "'Sachin vs Lata Civil War", was posted on Snapchat and Facebook on Thursday.
The video has the revered singer and the master blaster - both enacted by Mr Bhat using Snapchat's face-swap feature - "arguing" about who is the better player, Sachin or Virat Kohli. It was posted with this disclaimer by Mr Bhat: "I make such nonsense on my Snapchat."
By Sunday, the video had been widely shared and had triggered far more outrage than deserved by "nonsense".
Filmmaker Pehlaj Nihalani, the Censor Board Chief, told NDTV: "Tanmay Bhat is a repeat offender, he deserves no freedom. He has insulted the mother of music."
Social media is divided on whether it is funny or weak or downright offensive. Or none of the above.
I am 9 times winner of #BestComicActor. Have a great sense of humor. But This's NOT humor. #Disgusting&Disrespectful https://t.co/sTuTfbAOrU
- Anupam Kher (@AnupamPkher) May 29, 2016
Am absolutely shocked. Disrespect is not cool and neither is it funny. https://t.co/ymYPi9hxuv
- Riteish Deshmukh (@Riteishd) May 28, 2016
Absolutely.. Shocked n appalled!! Not amused @mangeshkarlata ji needs to be apologised to .. NOW !! @rasheshshah https://t.co/viTcXyBSI0
- Celina Jaitly (@CelinaJaitly) May 28, 2016
Watched the Tanmay Bhat video. The Sachin-Virat joke was okay, but the ones abt Lata Mangeshkar's aged face & abusing her was in poor taste.
— Mihir Bijur (@MihirBijur) May 29, 2016
It's okay to subject people to humour, even legends like Lata & Sachin. But please show me the humour in that Tanmay video. It's just abuse
— The Bad Doctor (@DOCTORATLARGE) May 29, 2016
Tanmay Bhat kept up a steady stream of tweets but refused to back down.
LOL, Twitter, jaha pe logo ke marne se pehle unke marne ki khabar trend hoti hai "tastleless" aur "crass" jaise words use kar rahi hai
— Tanmay Bhat (@thetanmay) May 30, 2016
When you guys write about me tomorrow can you please use my snapchat ID - it's thetanmay thanks
- Tanmay Bhat (@thetanmay) May 28, 2016
Roast wale din yaad aa gaye by god pic.twitter.com/VgEPkaZy9e
- Tanmay Bhat (@thetanmay) May 29, 2016
The group AIB is known for its irreverent comedy and was caught in a massive controversy over an expletive-laden roast of actors Ranveer Singh and Arjun Kapoor in December 2014.
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