The charity no longer needs the red-roofed station and has to spend £6,000 a year maintaining it, she claims.
The Welsh Assembly's environment minister, Jane Davidson, has refused to grant the charity permission to demolish it.
So now the building is being marketed in the hope that someone will take it on, possibly as a new visitor attraction.
"It is ridiculous that the RNLI is being burdened with this millstone," said South Pembrokeshire AM, Mrs Burns.
"The RNLI is funded entirely by contributions and the money needs to be spent on saving lives, not old buildings."
Mrs. Burns has just been given a tour and a briefing on what is Wales' fifth-busiest lifeboat station. "The new £6 million station and boat are extremely impressive and the team there are incredibly modest and unassuming about what they do," she remarked.
"The money they raise is needed for active duty, not for preserving historic monuments - that's someone else's job."