During a television show before the Gujarat elections, Mr Nirupam made derogatory remarks on air about Ms Irani.
Both politicians appeared on the show together and Mr Nirupam challenged Ms Irani's comments. "You used to charge money to perform dance shows on TV. Now you are an election analyst," he had said.
The BJP's youth and women's wings had staged a protest against Mr Nirupam in Mumbai.
The remark has also invited angry reactions online, with some pointing to Mr Nirupam's foray, for a week in 2008, into television through reality show Bigg Boss.