Councillors also decided the centre would stop taking plastics number 3,4,6 and 7, because there is no market for them at present.
Those plastics are only about 7 per cent of collected plastic, councillor and waste minimisation chairman Rob Vinsen said.
"It will barely be noticed by the public."
But he said the decision to stop accepting paper and cardboard would be noticed.
Councillors worried it would erode public confidence in recycling and give young people the wrong message.
Agreeing the centre would stop taking it was the most difficult decision of the council's August 3 meeting, mayor Hamish McDouall said.
Councillor Jenny Duncan said people had just made the transition to using paper bags rather than plastic bags, because paper is biodegradable.
"We actually want to encourage the use of fibre," she said. "We just need to find a way to recycle it."
The Whanganui centre collects about 1000 tonnes of fibre a year, and used to get up to $80,000 a year by selling it.
Recently its fibre has been bundled with fibre from other regions and marketed by Oji Fibre Solutions. Sometimes the centre gets money from the sale, but more often there is a deficit.
"Any delay in this [decision] is basically costing $15,000 a month. I appeal to councillors not to delay on this," Vinsen said.
The council could have decided the centre should charge for taking fibre but a report to councillors by its waste minimisation adviser Stuart Hylton said the public backlash would have been hard on staff.
Or the council could have continued to pay for the recycling out of its Waste Minimisation Fund. That would cost $180,000 to $200,000 for the 2020-21 financial year.
"We don't believe that's a wise use of the waste minimisation money," Vinsen said.
Ceasing to take fibre might push Oji into offering the centre a better deal, Vinsen said. It could also inspire innovative uses for fibre - but Hylton said any national solution could be three to five years away.
Waste fibre is recycled into second-grade fibre, and can be used for packaging. It can also be composted with other organic waste. But Hylton said composting efforts in Whanganui are "not that advanced".
There is one operator doing an in-vessel composting operation and the council is working with neighbouring local authorities to use fibre in an organic composting operation.
The only long-term solution to the waste conundrum is a circular economy with a product stewardship scheme, like the kind Government announced for plastics last week, Hylton said.
"It's about not using products that we have no end point use for."
A decision about when paper and cardboard drop off will cease will be made after the Whanganui Resource Recovery Centre Trust meets on August 11.