Incumbent Ivory Coast President Laurent Gbagbo had earlier demanded that all foreign peacekeepers leave the country.
His spokesperson accused UN and French troops of colluding with former rebels.
The UN and major powers have recognised Mr Gbagbo's rival, Alassane Ouattara, as the winner of the 28 November poll. Mr Gbagbo insists he has won.
Mr Ouattara is currently under UN protection at a hotel in Abidjan.
In a statement read out on national television on Saturday, government spokeswoman Jacqueline Oble said Mr Gbagbo had "requested the immediate departure of the Unoci [UN mission] and the French forces supporting it".
The UN force, she added, had "interfered seriously in the internal affairs of Ivory Coast".
Mr Ban responded with his own statement saying the UN mission "will fulfil its mandate and will continue to monitor and document any human rights violations, incitement to hatred and violence, or attacks on UN peacekeepers".
Earlier, the UN said one of its patrols had come under fire as it entered the mission compound in Abidjan, the country's main city.
On Friday, opposition supporters were arrested in Grand Bassam, about 30km (20 miles) east of Abidjan, and there are reports of several people being killed there.
A day earlier at least 20 people were killed in clashes between Mr Ouattara's supporters and the security forces.
The UN Security Council has warned that all sides will be held accountable under international law for any attacks against civilians.
The United Nations, the US, former colonial power France, and the African Union have all called on Mr Gbagbo to stand down.
French President Nicolas Sarkozy on Friday said he should quit by Sunday or face EU sanctions.
But Mr Gbagbo says the election was rigged by rebels who still hold the north after the civil war in 2002-03.
Mr Gbagbo was declared the winner by the Constitutional Council after it annulled votes in parts of the north.
The BBC's John James, in Abidjan, says tensions have been building between Mr Gbagbo and the UN mission since it decided to recognise Mr Ouattara as the winner of the election.
A peace deal signed by all sides gave the UN a role in certifying the election results.