A nervous wife bustles about the log cabin with wads of bills in her hand. She hears her husband coming, and dashes to the woodburning stove – shoving the money inside.
Husband enters, lights the stove. Loud argument ensues.
Husband departs, the camera follows him. He calls out in a loud voice “Gold and silver call to me, for I’ve got a wife who burns money!” He tosses his pickaxe in a grand gesture off to the side, it strikes a rock wall – and gold falls out.
“The Unsinkable Molly Brown” (1964)?
It’s in color, but parts of it are black & white. Here is part of the plot synopsis from Wikipedia which seems to match your description:
“Rescued from the Colorado River as an infant and raised by Seamus Tobin, tomboy Molly Brown is determined to find a wealthy man to marry. She journeys to Leadville, Colorado and is hired as a saloon singer by Christmas Morgan. After miner Johnny Brown renovates his cabin, the two wed, and he sells his claim in a silver mine for $300,000. Soon after the money Molly hid in the stove is accidentally burnt, Johnny discovers the richest gold vein in state history.”