Mantle is primarily a noun referring to (1) a loose, sleeveless coat or cloak, (2) something that covers, (3) the layer of the Earth between the crust and the core, and (4) the cerebral cortex. It also has a rarely used verb definition—to cover as with a mantle.
Mantel is easy; it refers to the ornamental shelf above a fireplace. It has no other definitions in today’s English.
Mantle is the correct spelling in the various forms of the phrases under the mantle and take/pick up/put on the mantle. The mantle in these expressions is a metaphorical cloak (see definition 1 above).
Located on the top of Liberty Hill, the home … features an open living room and dining room with a fireplace set in a marble mantel. [San Francisco Chronicle]
The resort’s furniture, the wine cellar and the carved mantel over the fireplace in the bar were made on-site in his woodworking shop. [Montreal Gazette]
Your opinions that keeping ashes in an urn on the mantel seemed “creepy,” and scattering them in public places “slightly ghoulish,” were very strong. [letter to Daily Pilot]