[eternauta3k]

There are old propane refrigerators which do exactly that (heat powered heat pumps).

Have you done the math to check whether it's more efficient?

[ip26]

A natural gas furnace is up to 98% efficient. A natural gas turbine is 40-60% efficient; a modern heat pump can have a COP of 3 or even 4, for 150-200% equivalent efficiency (ballpark).