Unpaid staff at Felbrigg Hall in Norfolk have been told by senior staff that they need to be welcoming to LGBTQ visitors.
Those who don't wish to wear the badge will instead be given roles without contact with visitors at the 17th century hall.
Peter Lynas from the Northern Ireland Evangelical Alliance questioned the idea on Premier's News Hour.
He said: "Christians are entitled to say 'look biblically same-sex practice is wrong but we want to welcome the community' and so some may find it difficult to wear a lanyard that's showing support.
"And I think it's worth noting as well that people complaining aren't saying it's on the basis of faith, they just find the whole idea unacceptable as a volunteer."
The move is part of the trust's Prejudice and Pride campaign marking 50 years since the decriminalisation of homosexuality.
Meanwhile, the trust has been accused of "outing" Robert Wyndham Ketton-Cremer, the late owner of the Jacobean hall, following a short film narrated last month by TV personality Stephen Fry.
Ketton-Cremer's relatives wrote to The Daily Telegraph complaining that the "intensely private" historian and poet, who died in 1969, shouldn't have been exposed as gay.
At least 10 of the mansion's volunteers are understood to have refused to wear the rainbow badges in protest of the trust's decision to reveal Ketton-Cremer's sexuality.
Listen to Peter Lynas speaking with Premier's Eno Adeogun:
