Your Chimney Sheep® should last for years. It will get sooty but it can be washed on a gentle wash. It might fray a bit at the sides if you have a brick flue but this can be trimmed with a pair of scissors.
However the time may come when you decide you need a new felt head, the handle and clamp can be removed, and the handle unscrewed. You can then order a replacement felt head to fit the plastic components. The old sooty felt head can be disposed of in a compost heap if you have one.