Mr Cunliffe said he would not appear on the show "at the moment", with some party members unhappy with Tamihere's comments.
Tamihere is a Labour Party member who has aspirations to return to politics. Mr Cunliffe said whether the controversy would affect his chances of selection as a candidate or on the list was a matter for the party.
Vodafone, Telecom, Countdown and Briscoes yesterday followed other advertisers out the door and announced they were suspending all RadioLive advertising.
ANZ and Yellow have already withdrawn ads from the station and Freeview and AA Insurance's campaigns will no longer air during the Willie and JT Show.
RadioLive yesterday announced on its noon bulletin - just before the start of the programme - that due to advertiser withdrawals, the pair's three hours on air would be ad-free.
During the show, the two hosts avoided any discussion of the Roast Busters or the ad pull-out. The closest they came to raising the subject was quickly ended.
"I'll talk to my erstwhile colleague shortly about these [indistinguishable] in terms of their commercial pulling. You know, the day that in a democracy, you get people using their, um ... anyway, um ... where are we going to now?" Tamihere said.
MediaWorks refused to comment on whether Jackson and Tamihere had been told not to talk about the Roast Busters or the advertising pull-out on air, or whether the company was considering their positions. A spokeswoman said the company did not discuss employment matters.
MediaWorks also turned down a request for an interview with managing director Sussan Turner or radio boss Belinda Mulgrew.
Tamihere could not be reached for comment last night.
Calls for the pair to step down have been growing all week, particularly online, and broadcaster Martin Devlin yesterday added his voice to those believing the presenters should resign or be sacked.
The presenters sparked outrage on Tuesday with their questioning of Amy, an 18-year-old friend of a Roast Busters victim, which many branded victim-blaming.
This was followed by a walk-out on Thursday's show after an angry on-air row with columnist Matthew Hooton, who said he would not return to the show. His fellow guest panellist, union boss Matt McCarten, has yet to say if he will be back.
APNZ