While we know there is no official political relationship between Iran and USA, I want to know about the last one. I do not know about unofficial meetings or phone calls which may happen.
As a consequence of the hostage crisis,
On April 7, 1980, Carter severed diplomatic relations between Iran and the United States and they have been frozen ever since.
US don't have an embassy in Iran, and Iran has no embassy in the US. Heads of states of both sides have never met in the last 40 years.
However, especially during the Obama administration and after Rohani's election, diplomatic talks have been staged and lead to the Iran Nuclear Deal. The highest level of these encouters included ministers of Foreign affairs, hence a governmental level.
You can see here an official meeting between ministers of Foreign affairs Mohammad Javad Zarif (Iran) and John Kerry (USA), held in Lausanne, 2015, and leading to the Joint Comprehensive Plan of Action, or Iran Nuclear Deal.
Befor Iranian Revolution, already mentionined in the answer by DJClayworth, Iran was one of the main US allies in the region as can be read in this wikipedia article about Iran–United States relations. If it´s only about face to face meeting, then the las such meeting seems to be 1977. I am tallking about The Iranian Shah, Mohammad Reza Pahlavi meeting with Alfred Atherton, William H. Sullivan, Cyrus Vance, President Jimmy Carter, and Zbigniew Brzezinski. To be more precise the meeting took place December 31, 1977 where Carter made his controversal New Year's Eve 1978 toast to the Shah. After that and untill the begin of the Iranian Revolution there were still contacts betwee Iranian and US officials, but I have no information if it were face to face meetings. Since the revolution started just a few days after that meeting 7 January 1978, it´s wery unlikely that any face to face meetings took place after that.
While 1977 was the last meeting where both Iran and USA were allies, the next and at the moment last meeting between the representants of the two governments took place 2015. The meeting was 14 January 2015 in Geneva during the negotiations about Iran nucler deal as can be read in this BBC article.
Kerry said that he met with Zarif also after that meeting, but at that time he was not representing the US government any more.