Before he quit, Rogers was due to lead the UK's negotiations to leave the EU.
Prime Minister Theresa May has said she wants to trigger Article 50 by the end of March 2017, paving the way for the process to commence.
Rogers' resignation was described by former Deputy Prime Minister Nick Clegg as a "body blow" to the Conservative government's plans.
It comes weeks after the EU's chief Brexit negotiator, Michel Barnier, warned Britain
that it may only have 18 months to strike an exit deal from the bloc.
Barnier said the process would need to be completed by October 2018 to allow for ratification by the 27 remaining member states within the two-year time scale prescribed by EU rules.
Rogers reportedly told ministers
in October that a new free trade agreement with the EU could take 10 years to negotiate. But the prime minister's office said that did not reflect the government's view.
He had been in his post since November 2013 and was not due to depart until October 2017. News of his early exit caused some to speculate that it will damage Britain's negotiating position.
The opposition Labour Party's former foreign secretary Margaret Beckett told the Press Association: "I think it's very worrying. If, as it appears, he is lost because he gave honest advice, that is not a good precedent.
"I think it is a crisis for the Prime Minister and for the Foreign Office. It is deeply alarming."
Clegg said in a statement issued by the Liberal Democrats: "The resignation of somebody as experienced as Sir Ivan Rogers is a body blow to the Government's Brexit plans."
Former Labour cabinet minister Peter Mandelson said in a statement issued by the pro-Europe Open Britain campaign: "In terms of knowledge and experience of the EU, Sir Ivan Rogers is second to none in Whitehall. His resignation is a serious loss for us in Brussels.
"I would not expect him to comment further but everyone knows that civil servants are being increasingly inhibited in offering objective opinion and advice to Ministers."
Charles Grant, director of the Centre for European Reform, wrote in a tweet that Rogers was one of very few people at the top of the British government who understand the EU, and said his resignation "makes a good deal on Brexit less likely."
But Nigel Farage, former leader of the pro-Brexit UK Independence Party, welcomed the resignation, adding: "The Foreign Office needs a complete clear out."
And Conservative former shadow chancellor and member of the Brexit committee Peter Lilley told the Press Association the timing of the resignation would be a "very minor" issue for the prime minister.
A government spokesperson said: "Sir Ivan Rogers has resigned a few months early as UK Permanent Representative to the European Union. Sir Ivan has taken this decision now to enable a successor to be appointed before the UK invokes Article 50 by the end of March. We are grateful for his work and commitment over the last three years."