Paul Chambers, 26, was found guilty of sending a menacing electronic communication at Doncaster Magistrates' Court in May.
His tweet, sent in the early hours of January 6, said: "Crap! Robin Hood Airport is closed. You've got a week and a bit to get your sh*t together, otherwise I'm blowing the airport sky high!"
Chambers claimed he sent the message to his 600 "followers" on Twitter in a moment of frustration after nearby Robin Hood Airport was closed by snow.
But a district judge ruled the Tweet was "of a menacing nature in the context of the times in which we live", finding Chambers guilty under section 127 of the Communications Act 2003. He was ordered to pay a 385 fine, a 15 victims surcharge and 600 costs
Chambers, from Doncaster, South Yorkshire, launched a battle to clear his name over the Tweet, arguing that it was "an ill-conceived attempt at humour" rather than a serious threat. His appeal was rejected by Judge Jacqueline Davies at Doncaster Crown Court this month.
The decision was met with dismay by freedom of speech campaigners, including comedian Stephen Fry, who offered to pay Chambers' fine and legal costs, amounting to nearly 3,600.
Chambers, who now lives in Northern Ireland, said on Twitter that his High Court challenge was "probably to the detriment of my mental wellbeing" but that he was "appealing the decision as best I can".
He added: "Thanks for all the support. It was very tempting to draw a line to be honest, took a long time to weigh it up. I feel like I'm living in a goldfish bowl." Chambers' decision to continue the appeal against his conviction has encouraged others to speak out in support.
No date has been set for Chambers' High Court appearance, but it is likely to be early next year.