Paul Chambers, 27, from Doncaster, who now lives in Northern Ireland, was convicted in May 2011 of sending a "menacing electronic communication".
He claimed it was a joke and wants his conviction and sentence quashed.
Comedian Stephen Fry, who was at the High Court hearing, said it was "very important" for freedom of speech.
The judgement was reserved until a later date.
Mr Chambers said he had sent the tweet, which contained swear words, to his 600 followers in a moment of frustration after Robin Hood Airport in South Yorkshire was closed by snow in January 2010.
He said he had never thought anyone would take his "silly joke" seriously.
The message Chambers tweeted stated: "Crap! Robin Hood airport is closed. You've got a week and a bit to get your shit together, otherwise I'm blowing the airport sky high!!"
By Dominic CascianiHome affairs correspondent
Paul Chambers' QC told the High Court that you can't define exactly a "menacing threat" - but "you know it when you see it."
And this is the question at the heart of the appeal: Did those who read the Tweet regard it as a joke or a bomb threat?
His lawyers argue the context of the words are very important and that the judges should take into account where and how they appeared on his Twitter timeline. Whether the judges will draw a distinction between offensive messages on social networks and those sent by other means remains to be seen.
The case to uphold the conviction is simple: It doesn't matter if Mr Chambers' friends chortled as they read his tweet as a joke. It became a crime because anyone at all, of reasonable state of mind, could regard it as a genuine threat. And that's the argument he has to defeat.
He was found guilty by Doncaster magistrates in May 2010, fined ??385 and ordered to pay ??600 costs.
An appeal was dismissed in November 2010 with a Crown court judge stating that the electronic communication was "clearly menacing" and that airport staff were sufficiently concerned to report it.
His lawyers have claimed he was the victim of a legal "steamroller" that threatened to make the law look silly and that the Crown court erred in law and in common sense.
John Cooper QC, representing Chambers at the High Court, argued that even if the message was a threat, it could not be defined as menacing or criminal.
He told the High Court hearing: "We don't say it's a good joke but he shouldn't have been convicted over a bad one.
"At worst, the tweet was offensive."
Mr Cooper said for a message to be considered menacing, the person sending the message must intend to threaten the recipient and it has to contain a credible threat.
During the hearing, Mr Cooper was challenged by one of the judges on whether there was a genuine freedom of speech argument in the case.
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The QC said that freedom of speech was not just there to protect the righteous but to protect the unpleasant too.
Among Mr Chambers' supporters at the hearing were Stephen Fry and comedian Al Murray.
Mr Murray said he was there because he had found the conviction "monstrously unjust".
He said: "[Mr Chambers] made a throwaway remark and the law has lumbered into horrendous action.
"It astounds me that anyone found this threatening."
Judgement has been reserved in the appeal of a man who was found guilty of posting a comment on Twitter threatening to blow up an airport.