Mere threats roiled oil markets in late September; actual hostilities would certainly drive up prices for longer.
How high? That’s hard to say. After Iran’s 1979 Islamic revolution and the start of the Iran-Iraq War the next year oil prices doubled. In the months after Iraq invaded Kuwait in August 1990, oil prices doubled again, helping to tip the US into recession, though the market cooled almost as fast as it had heated up. If hostilities erupted and Iran sought to close the Strait of Hormuz, prices could easily double in the short run.
Even in good times, oil spikes threaten recession. With the US in a fragile recovery, Europe reeling from its debt woes, and signs of slowing growth in China, the global economy is about as poorly placed as possible to weather an oil shock.