WASHINGTON—Tensions between the U.S. and Iran, rather than easing as a result of July’s nuclear accord, are increasing over a wide spectrum of issues tied to the broader Middle East security landscape and to domestic Iranian politics, current and former U.S. officials say.

Just in the past two days, Iran has test-fired a ballistic missile and announced the conviction of American journalist Jason Rezaian, fueling suspicions the historic nuclear agreement has allowed Tehran’s Islamist clerics to step up their long-held anti-U.S....