TEHRAN—Iran’s falling currency is causing many Iranians to cancel vacations, business trips, college studies and medical treatment abroad, a retreat in consumer spending that is weighing on the country’s troubled economy.

The Iranian rial has lost almost three-quarters of its value against the U.S. dollar since January to hit a low of about 140,000 to the dollar this month. That means goods and services priced in dollars are far more costly, including aviation fuel, making travel prohibitively expensive for many middle class...