Aditya, son of the party president Uddhav Thackeray and Yuva Sena chief, met Governor Ch Vidyasagar Rao at the Raj Bhavan to demand action against Mr Tawde, a senior BJP minister, and Mr Deshmukh.
The Yuva Sena leader's demand for Mr Tawde and Mr Deshmukh's resignation came even as MU announced four lecture-free days to expedite the online assessment process.
"I urge you to take their resignation for their absolute failure to conduct the basic work of the Mumbai University, taking it to a pit that it has never gone down to before in its over 100 years old history," Aaditya said in a letter submitted to the governor.
He later tweeted, "We've protested against MU's faults for the last seven years and got it corrected. Never before was it so bad and chaotic as today."
"Are the Education Minister and the Vice Chancellor understanding the needs of the MU and education or must they resign?" he added.
"MU is now without a Pro Vice Chancellor, registrar and has an ad hoc controller of examinations and is in the middle of a mess," he said.
"With hurried assessment, what is the quality of assessment? The MU must waive off Rs 600 charged for revaluation," Aaditya said.
"With only a few lakh papers corrected by now, will all assessment meet the 31st (July) deadline? Will marksheets be given too?" he said.
"An enquiry must happen into why was online assessment opted for? How was the tender process? Since it has failed so badly, is it a scam?" he said.
"I raised the issue of the absolute mismanagement of the Mumbai University by the VC," Aaditya said.