Homosexuality is illegal in Qatar, where same-sex relationships can be punishable by the death sentence.
Qatar has said all fans will be welcomed "without discrimination" but businessman Lawrence Barton said his venues would not be showing games.
"Our community matters more than making money," he said.
Mr Barton, who is also director of Birmingham Pride, said his venues had been packed when they had shown games for previous tournaments.
"It's usually a fantastic success, really well attended but I'm afraid there is no way I could ever consider profiting from knowing that that is the situation in Qatar with our community," he added.
"As a visitor [to Qatar] I could be arrested and put in prison for simply being gay."
Elsewhere, Birmingham comedian Joe Lycett issued an ultimatum to Qatar World Cup ambassador David Beckham.
He promised to donate £10,000 of his own money to charity but only if Beckham ended his "reported £10million" deal with Qatar.
If he refused, Lycett said he would shred the £10,000 just before the opening ceremony of the World Cup.
David Beckham has been contacted by the BBC for comment.
There are widespread concerns about how LGBT people are treated in Qatar and Beckham has been urged to speak out on the criminalisation of same-sex relations in the country.
Sam Timms, founder of Aston Villa's LGBT supporters network, said he "took his hat off" to Lycett for highlighting the issue.
People need to stand up because the community has been treated as a "political tennis ball", he added.
"It's crazy to think you as your authentic self are restricted to go somewhere in the world," Mr Timms said.
Qatar's World Cup organisers have stated "everyone is welcome" to visit the country to watch the football matches.
“I'm afraid there is no way I could ever consider profiting from knowing that that is the situation in Qatar with our community” Lawrence Barton
