No. Chador (a piece of clothing that covers the whole body from head to ankles) is not compulsory in Iran. You can see religious people wearing black Chador on the streets, or colorful ones in the shrines to spice up their pictures, but there is no need for that and it’s mandatory.
In some holy shrines, like Imam Reza in Mashhad, Hizrat Masumeh in Qom, or Shah Cheragh in Shiraz wearing Chador is necessary. In this case, one will be given to you at the entrance, so you don’t have to buy one for your trip.