Campbell, the former spokesman for Tony Blair and the inspiration behind the character Malcolm Tucker in Iannucci's BBC sitcom The Thick Of It, mocked the writer for accepting the honour.
Yesterday on Twitter, Iannucci declared it was "the most humble day of my life", referring to his OBE. Campbell responded by accusing Iannucci of joining those he had become famous for parodying.
Campbell tweeted: "So @AIannucci OBE joins the establishment he claims to deride, Malcolm Tucker and I do not approve of honours system," sparking a series of cantankerous exchanges with the writer, who once penned a spoof column posing as the former New Labour spin doctor.
The comedian responded by comparing his alleged offence to the Blair government's decision to invade Iraq, tweeting: "It's probably more establishment to order your army to march into other countries for no reason, swings and roundabouts."
Iannucci also denied his OBE would affect his comedy. He said: "Does Chris Hoy cycle less well after being honoured? Is there a suggestion he has sold out? My comedy is not about judging where people come from, or what they are called or what school they went to. I don't care, it's about what they do."
And he said: "If you accept any awards or prizes then why not this one? I have accepted honorary degrees too and it seems to me bad manners not to [accept]."
Broadcast journalists Andrew Neil and Samira Ahmed also joined the debate, with Neil asking: "Why would somebody who constantly parades their anti-establishment credentials accept such a bauble?"
Speaking later, Campbell said that although he admired Iannucci's work, he couldn't resist asking the question because of his own opposition to the honours system.
"I have always had a bit of a thing about the honours system - People may ask why we did not change it while we had the chance, but Tony Blair did not see it as a priority. He also felt it was something that made a lot of people very happy."
Campbell said it seemed strange that someone who invented Malcolm Tucker should start "spinning his own answers to these questions".
He said: "[Iannucci] is definitely popular and rightly so. But if you have made your name by making fun of the establishment, which he has done very cleverly, then it weakens it if you accept an honour."
l Meanwhile, in the latest part of Campbell's diary serialisation, he claims Rupert Murdoch called Tony Blair, urging him not to delay the invasion of Iraq.
In his book, Campbell says there were three phone calls in March 2003, a week before a crucial vote on Iraq in the House of Commons. News International dismissed the claims as "complete rubbish".
Murdoch previously told the Leveson Inquiry he "never asked a prime minister for anything".
In May, Blair, who is godfather to one of Murdoch's children, defended his friendship with the media mogul saying they had a "working relationship".