Not sure how BLO will do. I like to finish my trivets with chopping board oil, like this Amazon.com: John Boos Block All-Natural Board Cream for Butcher Blocks and Cutting Boards, 5 oz - Wood Conditioner & Sealer for Maintenance and Protection, Safe Formula with Beeswax & Mineral Oil : Health & Household. There are many other brands of similar stuff.
It consists of beeswax and mineral oil. The mineral oil evaporates off, leaving the beeswax. When the trivet gets hot, the beeswax melts and absorbs into the wood.