Like other answers say... Your problem is that the act of forcibly pumping energy one place to another, beyond what nature itself would do, takes additional energy.
So for example, moving the warmth from inside your fridge to outside it, costs extra energy. Informally (not literally to scale), you can think that the fridge interior loses 100 units of heat energy, so it does cool down. But the outside - the room and ultimately the planet as a whole - gains 110 units of heat energy: 100 units moved from inside the fridge, but also an extra 10 units from the work the pump must do, to move that 100 units from inside to outside. Overall the planet heats up.
Your lasers will do the same. They will send 100 units of heat energy into space, but to do so will require an extra 10,000 units of heat into the room, or the planet generally. (Not literally to scale). Big lasers take a lot of energy to fire up.
Similarly with anything collecting low grade heat and concentrating it as high density heat, or collecting solar energy to pump heat around. These all pretty much will add more heat to the planet than they remove, as they move heat round.
If you want to remove heat from earth, the best ways are 1) stop adding heat, or be more energy efficient and reduce the energy you use on earth, 2) make it easier for heat to escape (reduce heat-retaining mechanisms: CO2, methane, etc), 3) move energy consumption into space.