The club were the only one of 62 clubs who refused to accept a 10 per cent reduction in fees paid for coverage rights.
Many Wednesday fans have been left 'embarrassed' by the club's decision and have accused owner Dejphon Chansiri of 'money-grabbing'.
Fans wanting to listen to Wednesday's matches will now have to subscribe to the club's iFollow service for £4.49 a month.
Wednesday explained that the terms offered by the BBC were 'significantly less' than those within the year-by-year contract they've signed every summer.
But, Owls fan Lee Hicklin has been left unimpressed with this explanation and has launched a petition urging the club to reverse their decision.
He explained: "While it is clear that the club want to make a stand regarding the 10% reduction in fees, and that the club is trying to do more to meet Profit & Sustainability guidelines (the club has reacted by suggesting fans instead subscribe to its own iFollow service for £4.49 a month), the severance of the deal for BBC Radio Sheffield to cover games live comes as a huge blow to a large portion of the Wednesday family.
"Many older fans unfamiliar with the internet will not be able to make the switch to iFollow, nor will those who have already committed so much to the club financially in recent years.
"Financially this decision may make sense and save the club thousands of pounds, but is that worth the negative ramifications on the Wednesday fanbase? Or the negative PR received?
"We urge the club to reverse this decision and let the BBC continue covering Wednesday's away games live on Radio Sheffield."
Almost 100 fans have already signed the petition in less than an hour and many more have called on the club to rethink their petition.
Iain Chapman tweeted: "My father in law suffers from Dementia, used to go 2 away games but can't any more, my wife has to take him to all home games. Been goin for 70 years! SWFC have deprived him of one of the few pleasures he has in life. Thoughtless!"
Sheffield United fan Andrew Clark wrote: "Putting club differences aside, the good people of Sheffield deserve to be allowed to listen to their team play. There are, as some have said, vulnerable and disabled folk relying on the service, people overseas just wanting to be a part of the match day atmosphere. Reconsider!!"
Keiron Wilcock tweeted: "Twitter updates & sky sports news desk is not a patch on listening to John Pearson giving us live commentary. Lucky for me I go to most away games. Feel gutted for the older generation. It needs sorting."
Sheffield United, Rotherham, Barnsley and Doncaster matches will all continue to be covered live by BBC Radio Sheffield.
A radio station spokesman said: "We are disappointed Sheffield Wednesday hasn’t accepted the new commentary deal. "We have a long history with the club and have enjoyed bringing fans live commentary of The Owls’ home and away fixtures.
"We’re sad we will no longer be able to provide this service.
"We want to work with the club in the future and will continue to provide the latest news about Sheffield Wednesday to our listeners.
"We will also continue to have commentaries from Sheffield United, Rotherham United, Doncaster Rovers and Barnsley."
BBC Radio Sheffield has provided live commentary on Owls games every season since 2000. It will continue to provide score flashes from games along with pre and post-match summaries.
To view the petition, click here.