The threats came after an exchange of strikes between the US and Iran earlier this week, sparked by attacks on three commercial tankers, purportedly by Iran.
The exchanges have raised tensions between the two sides, prompting Trump to declare the ceasefire over, though he said talks aimed at ending the war would continue.
Ali Khamenei was buried in his home city of Mashhad on Friday after nearly a week of funeral ceremonies.
His son has not been seen in public since before the war, and is rumoured to have been disfigured in the strike which killed his father.
But his statement on Saturday was uncompromising.
“We pledge to avenge the blood of the martyred leader and all the martyrs of these two wars from the criminal and disgraced killers,” he was quoted as saying.
“Vengeance is the will of our nation and must inevitably be carried out.
“The matter depends neither on my personal existence nor on that of other officials. Whether we are present or not, it will come to pass.”
Those responsible “will take the wish for a peaceful death in their beds to their graves”, he added.
There have been open calls for Trump’s assassination at the funeral.