There's no instance I can recall of a president shutting down something comparable to Infowars. But there is an instance of a presidential candidate doing so.
In 1964 Barry Goldwater ran for president as the Republican candidate. He was far to the right on many issues, and many of his supporters were even farther to the right. He was opposed to civil rights laws and the civil rights movement, believed in a communist conspiracy to take over our country and the world, and suggested in an interview that if he were president he'd use nuclear weapons in Vietnam.
Many on the left feared Goldwater might actually be crazy enough to do it. And a couple of months before the 1964 presidential election,
Fact magazine published an issue devoted to saying just that. The issue consisted of a 20-page article from publisher Ralph Ginzburg quoting many of the crazy things Goldwater had said over the years and a 42 page article quoting the opinions of more than 100 psychiatrists regarding Goldwater's mental condition (most of whom shared Fact's view that he was not of sound enough mind to be a good president).
Goldwater sued. In 1967 he won $1 in compensatory damages and $75,000 in punitive damages, which I believe was a key reason why Ginzburg stopped publishing Fact magazine later that year.
Fact magazine specialized in somewhat sensational articles on social and political issues of its time (with a little sex thrown in). Here's an
online item I was able to locate which gives a little sense of what the articles were like. Fact was not nearly as into crazy conspiracy theories and dishonest misinformation as Infowars, but both strike me as similar in being sensationalistic outlets printing stories designed to attack people they dislike rather than news outlets printing news stories.
I was (and still am) a fan of Fact magazine, which printed a number of valuable articles the mainstream media of the time wouldn't touch, but I think Goldwater was within his rights to file suit against Fact and I do not think his action in doing so is equivalent to what Trump is threatening. If anyone (citizen, politician, or even president) were to take similar action against Infowars and as a result Infowars ceased to be I would not have a first amendment problem with that.