Some heat pumps use underground pipes (and those units have a great advantage in avoiding the efficiency loss due to extreme temperatures), however they're much more expensive to install, so the vast majority of heat pumps are air-based, same as any traditional air conditioner you've ever used.
As for whether ground-based heat pumps would cool the Earth's crust, the answer is no (in any measurable sense). Consider that heat pumps are used for both heating in winter and cooling in summer, so in a temperate climate you're just as likely to put as much heat into the ground (in summertime) as you extract from the ground (in wintertime).
As for whether ground-based heat pumps would cool the Earth's crust, the answer is no (in any measurable sense). Consider that heat pumps are used for both heating in winter and cooling in summer, so in a temperate climate you're just as likely to put as much heat into the ground (in summertime) as you extract from the ground (in wintertime).