The scale of the problem
An open fireplace with a leaky throat damper is a permanent opening between your living room and the outdoors. Even closed, a warped steel damper on a steel frame seals only approximately.
On an unused fireplace, that gap runs all winter for no benefit whatsoever.
Top-sealing dampers
A top-mounted damper seals at the top of the flue with a silicone gasket and is operated by a cable down to the firebox. It seals far better than a throat damper and keeps the whole flue out of the weather.
Keeping the flue warmer and drier has a secondary benefit: less condensation, less creosote and fewer odours in summer.
Glass doors and chimney balloons
Glass doors close the opening when the fireplace is cold and contain sparks when it is not. For a fireplace that is genuinely never used, an inflatable chimney balloon below the damper is an inexpensive seasonal seal.
Balloons need a visible reminder tag. Lighting a fire with one installed is exactly the mistake it sounds like.
The permanent fix
If you want the fireplace to contribute heat rather than lose it, a sealed insert with a properly sized liner is the answer. It controls combustion air and delivers heat to the room instead of the flue.
If you never intend to use it, sealing the flue properly and capping the chimney is the honest and cheaper decision.