A video on the site is headlined "Trashing a toy Trump: dynamite, acid, flamethrower, molten metal, thrown from a plane and more".
The prime minister's office told 2GB host Ben Fordham the staff member had been "suspended indefinitely" for what they deemed "inappropriate" conduct.
"You would think that because this staff member is working for the PM's office, she would be wise enough not to post something like this," Fordham said.
"We know the relationship is already on shaky ground ... a staff member in his office posting anti-Trump propaganda is probably not the smartest move she's ever made."
He said the employee has accepted the posts were inappropriate.
The news follows reports out of Washington that a phone call between Mr Turnbull and Mr Trump over the resettlement of refugees on Manus Island and Nauru turned ugly.
It was reported by The Washington Post that Mr Trump called the deal "the worst deal ever" and told his Australian counterpart that he would get "killed" politically for it.
Mr Trump later tweeted that the deal was "dumb" and that he would study it, but his press secretary confirmed the US would not pull out of its pledge to take the 1250 men women and children from places including Pakistan, Iraq, Iran, Bangladesh and Sri Lanka.
News.com.au has attempted to contact the prime minister's office.
