They were his first remarks since Sunday's deadly US strikes on the Hashed al-Shaabi paramilitary network in western Iraq.
The United States carried out the raids in retaliation for rocket fire that killed an American civilian contractor at a base in northern Iraq on Friday.
In a tweet on Tuesday, Trump accused Iran of "orchestrating" that day's storming of the US embassy in Baghdad by protesters angry at the American air strikes.
Trump said: "Iran will be held fully responsible for lives lost, or damage incurred, at any of our facilities. They will pay a very BIG PRICE! This is not a Warning, it is a Threat."
In response, Khamenei retweeted the post and said: "That guy has tweeted that we (the United States) see Iran responsible for the events in Baghdad and we will respond to Iran.
"First of all, you can't do a damn thing! This has nothing to do with Iran," the supreme leader in his televised address.