The Taleban representative, Nurullah Zadran, told the BBC he was complying with the request.
In the western city of Herat, 80,000 refugees are living in tent camps and last month at least 150 people died of hunger and cold.
Tens of thousands of other refugees have converged on Mazar-e-Sharif and Baghlan in the north.
But aid agencies have not had enough food to reach all needy areas.
Mr Oshima said: "If international assistance does not arrive in greater quantity and with most faith, the situation may become very serious indeed."
Closure demand
The ruling Taleban movement says its office in New York received a formal letter from the United States Government on Tuesday ordering the closure of its mission there.
The Taleban representative, Nurullah Zadran, told the BBC he was complying with the request.
As Mr Oshima's mission continues, the UN senior representative to Afghanistan, Francesc Vendrell, says he still hopes a formula can be agreed with the US Government to permit a Taleban representative to remain in New York.
The Taleban said earlier that they would close down the UN political office in Kabul, in retaliation for the loss of their New York base. New arrivals
Mr Oshima's visit came as a jet carrying 7,500 quilted blankets and 300 cold-weather tents arrived in Herat from Norway.
Another plane with US relief supplies is expected on Wednesday, the second from the United States in as many weeks.
Last week Pakistan said it was rushing 5,000 tonnes of rice, 5,000 tents and 50,000 blankets and quilts to Afghanistan.
UN co-ordinator Hans-Christian Poulsen, said: "For the time being we have tents and blankets for the present population but there are constantly new people arriving.
"The situation is better for the refugees in and around Herat but outside in the districts it is as bad as ever - people are very short of food."
Animals eaten
In the main camp, Maslakh, two families of around six members each are sharing UN tents, which provide minimal protection from temperatures well below freezing at night.
Mohammad Ibrahim, 30, said he travelled for four days with 21
families from the same remote village to reach the camps around
Herat.
"We ran out of food and we had nothing so we were forced to leave," he said, adding that his harvest had completely failed and they had eaten their animals.
Drought, catastrophic crop failure and the continuing civil war between the Taleban and opposition forces in the north east means that a quarter of the population of the capital, Kabul, receives subsidised bread.
But the UN has been finding it hard to raise aid money.