Mr Miliband is the shadow climate change secretary, has been MP for Doncaster North since 2005 and is standing as a candidate at the general election.
He led the Labour Party from 2010 until after the 2015 general election.
No details of the incident involving Donaldson were outlined in court.
The Recorder of Sheffield, Judge Jeremy Richardson KC, told the defendant: “This is a serious matter, for very obvious reasons.
“It’s always a serious matter to threaten to kill anyone. To threaten to kill a member of parliament in the current situation, given the relatively recent history, is a serious crime.”
Donaldson was remanded in custody.
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