Iranian President Mahmoud Ahmadinejad described Israel as a "cancerous tumor" last week during a national day of protest. Iranian Supreme Leader Ali Khamanei said Israel would "disappear."
A statement attributed to Ban said national leaders should use their influence to lower, not escalate, regional tensions.
The statement said Ban was "dismayed by the remarks threatening Israel's existence attributed over the last two days to the supreme leader and the president of the Islamic republic of Iran."
Iran designated Tuesday as National Defense Industry Day with ceremonies marking technological advances in the country's military arsenal.
Ahmadinejad was quoted by state-funded broadcaster Press TV as saying military and defense technology were paramount for his country's self-defense.
"In order to be deterrent, the defense industry should take advantage of the most advanced sciences and fortunately, the sets of equipment unveiled today are among the best in the world," he said.
An Iranian defense official said Monday the military was able to deliver "a crushing response to any aggressors."