Iran’s costly efforts to project power beyond its borders in the wider Middle East are exacting a political price at home, with arrests and deaths multiplying as antigovernment demonstrations persisted into a sixth day Tuesday.

The billions of dollars Iran spends on foreign conflicts have been a focal point of protester anger at a time when domestic inflation and unemployment are in double digits. Crowds chanting “Leave Syria, think of us!” are seeking to force Tehran to reassess a cornerstone of its foreign policy: the use...