The President tweeted anyone doing business with Iran will not be doing business with the United States.
Some re-imposed sanctions took effect overnight and tougher ones relating to oil exports will begin in November.
Iran’s President Hassan Ruhani said the measures were psychological warfare, which aimed to “sow division among Iranians.
The sanctions follow the US withdrawal from the Joint Comprehensive Plan of Action, also known as the Iran Nuclear Deal, earlier this year.
The deal, negotiated during the Presidency of Barack Obama, saw Iran limit its controversial nuclear activities in exchange for sanctions relief.