Leave.EU chairman Arron Banks welcomed the departure of a man he described as "far too much of a pessimist and yet another of the establishment's pro-EU old guard".
Rogers, who was not due to leave his position until November, was appointed to Brussels by David Cameron in November 2013 and is one of the UK's most experienced diplomats on European Union affairs.
May has said she intends to trigger Article 50, the formal process of leaving the European Union and beginning negotiations with Brussels, by the end of March.
Politicians, civil servants and commentators pointed out that Sir Ivan Rogers' expertise in Brussels would be very hard to replace, as the exit talks loom.
Last month, it was revealed that Rogers had privately advised ministers that the complexity of Brexit negotiations meant that talks could stretch into the mid-2020s, rather than the two years touted by the government. "He was representing what others are saying to him", said a source.
"Macpherson, who was at the heart of government for over a decade, suggested other key civil servants with European Union knowledge were "out of the loop" before finishing with the hashtag "#amateurism".
Leave campaigners at the time accused the diplomat of "gloomy pessimism". He has at least done the honourable thing in resigning,"he said. Good - time for some optimism!"
Hilary Benn, a Labour MP and the chair of the Brexit committee in the United Kingdom parliament, told BBC radio Rogers' departure is "not a good thing" for Britain's position.
"Our negotiation as a whole will go nowhere if ministers are going to delude themselves about the vast difficulty and challenges Britain faces in implementing the referendum decision", Mandelson said. "And having a handover in the middle of that, depending on when exactly he goes, is not ideal".
"I think it would be appropriate if a lot more people in that position, British ambassadors, left".
Senior Irish officials in regular contact with British counterparts, who spoke to The Irish Times on condition of anonymity, said they were not surprised by the departure of Mr Rogers.
"She should welcome it with open arms and put a firm Brexiteer in the position. The world has changed". Michael Heaver of UKIP, the right wing Euro-skeptic party, tweeted that May should "get someone who believes in Brexit".