This tool at Smith’s Tavern is a candle snuffer.
Candle wicks were made from twisted threads of flax, cotton, or hemp and didn't burn nearly as well as modern wicks. Trimming the wick to get rid of "candle snuffs" was an important part of keeping a candle burning well. If not attended to, the candle could get too hot, melt too much fat and send it streaming wastefully and messily down the sides - known as guttering. Smoking and excessive smell could also be improved by careful trimming.
Candle snuffers were not primarily for extinguishing the candle. Snuffers were like scissors (or nippers) for cutting off excess sooty thread. A sharp point was useful for spearing any scraps of burnt wick that fell into the hot tallow. The snuffers often had a box to catch those clipped threads - the "snuffs". Snuffers were sometimes called snuffer boxes or box snuffers.