Users accessing the OKCupid site through Firefox are currently shown a full-page message urging them to use alternative browsers.
It claims Mozilla CEO Brendan Eich is "an opponent of equal rights for gay couples", referring to his support for the campaign backing California's Proposition 8, a 2008 initiative that banned same-sex marriage in the state.
Octopus Group managing director Jon Lonsdale said: "It’s a live example of how leaders are, in themselves, a major part of what makes up any brand. Tech brands like Mozilla that champion emotive phrases like 'doing good', or even Google with 'don’t be evil', are perhaps easier targets than most for those with an agenda. And as we all know now, the social media universe latches on to any hint of irony and perceived double standards in business like a limpet."
Mozilla responded that OKCupid had not checked all the facts and the company "supports equality for all, including marriage equality for LGBT couples".