Heat pumps do work, but only within the laws of thermodynamics. So we can put say a 1kW electric radiator in our room, and get exactly 1kW of heating from it. Or we can use that 1 kW to run a heat pump, in which case we might get something like 3kW worth of heating in our room, for an input of 1 kW of electricity from the mains and 2 kW of cooling the great outdoors. But in the long run, all of the heat in our room will escape outdoors, and either way we will have used 1kW times the number of hours we ran either system of actual power that had to be generated somewhere. So no, the planet will not be frozen down, but we might be able to get by with less power used for heating. The actual output from the heat pump will depend on how cold it is at the outside end, the above example would be typical enough for where I live, where the air temperature outside rarely drops below freezing, but you will get less in a very cold climate. Heat pumps are pretty good but TANSTAAFL still applies. There is a practical limit to how cold the outside end can be allowed to get, since it has to be able to provide energy to evaporate the working fluid. Allowing it to be covered in ice will not help the heat to flow in. (Remember that even freezing point is a long way above absolute zero, so there is still heat available at freezing point, provided it can flow to your outside end.)
Pumping heat to a high temperature to generate power comes under the same general classification as pulling yourself up by your bootstraps. An ideal heat engine running off such a heat pump would generate just enough power to run an ideal heat pump, with no heat or mechanical energy able to be spared for actual use. Of course ideal heat pumps and heat engines are not actually possible.
There is an idea for how to extract energy from heat in the oceans, it would rely on the difference in temperature between the surface, especially in the tropics, and the deep ocean. But there is only a small temperature difference to work with, and done on a large scale it might interfere with ocean currents. This should be a particular concern to you guys in the UK, since the only reason the place is inhabitable is because of the Gulf stream.
John