Can we cool Earth by shooting powerful lasers into space? In a sense, the climate change discussion revolves around the unwanted warming of the earth's atmosphere as a whole.
It seems a bit too obvious to be true, but could we cool the atmosphere by simply shooting that unwanted energy somewhere else?
Energy might be collected from remote expanses where it would otherwise be somewhat pointless to harvest it due to lack of habitability and resulting anticipated losses due to transmission (ocean surface, ???)
If so, what would be a good place to shoot it?
 A: 
... could we cool the atmosphere by simply shooting that unwanted energy somewhere else?

Yes, but only by making our problems worse than they are already. A heat pump removes unwanted heat (that is how a refrigerator works) but it takes energy to make a heat pump work. Either this energy comes from non-renewable resources - which just generates more heat and makes our energy problems worse - or it comes from renewable resources such as solar power. But using energy from renewable resources to export heat that you have generated by consuming non-renewable resources makes no sense - just replace the non-renewable resources with the renewable resources in the first place.
Here is an analogy - you have lit a fire in your room to run a steam engine which generates all your electricity. But the fire is making the room too hot and you want to run an air conditioning unit to cool the room down. You could build a bigger fire to generate more electricity to run the air conditioning unit - but that makes your problem worse. Or you could install solar panels to generate electricity to power the air conditioning unit - but if you have solar panels why do you still need your fire and steam engine ?
A: 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.
A: Cheap and effective method: We should paint roofs and streets white.
Besides, a small part of the Sahara would currently be enough to supply humans with electricity from solar cells. On the rest of the Sahara there could be solar cells to power your lasers. But this is only possible if the power loss of cells and lasers that escapes into the environment is less than the part of the sunlight that does not leave the earth as reflected radiation nad heats the earth.
A: Your idea to cool the Earth by shooting photons off into space is actually what already happens now!  However, instead of a laser, the Earth cools itself by blackbody radiation.  The Earth radiates a $\mathrm{\approx 300~^\circ K}$ (room temperature) black body spectrum with a peak at $\mathrm{kT=\frac{1}{40}eV}$ which is just a little redder than an infrared laser.  The Earth also receives radiation from the Sun and a $\mathrm{3 ~^\circ K}$ blackbody spectrum from the cosmic microwave background (CMBR).  Overall the Earth is in equilibrium (its temperature is constant) so it radiates as much power as it receives.
The total power radiated by the Earth is:
$$\mathrm{Power=4\pi(R_{earth})^2\sigma T^4=2.3\times 10^5\quad Terawatts}$$
It would take some really big CW lasers to compete with this!!
A: Ambient heat is low grade energy. You can't effectively harness it to do work (which is a transfer of energy. i.e. powering laser which itself is a machine to transfer energy) because it has nowhere to flow effectively. If you could, we wouldn't have energy problems. At all. You could then harness waste heat from machines and make perpetual motion machines (or nearly so), or easily make it flow on its own somewhere else if there is too much (which was the goal of a fancy laser to begin with). In reality, it's like trying to power a dam or drain water away when the elevation everywhere is the same.
You would need something with the same fundamental concept of  a refrigerator which uses an input of high-grade energy (electricity) to do work. That work is shoving low-grade energy from one place to another because the low-grade energy doesn't want to flow anywhere on its own. The laser in this case acts like an active heatsink to shunt energy away once it's collected.
The catch is that this consumes good energy you have which might be scarce, and in itself produces waste heat when used too. So you also need a mechanism to get that new waste heat into the laser or if you can't then some way to move that heat away without the laser (which was your original problem) lest you have a net heat build-up.  It's almost a Catch-22.
Not impossible but tough. You would almost need a space elevator of a refrigerator to shunt heat up into machinery in space where you could do something to dissipate where the waste heat from the process could also be dissipated without immediately ending back on Earth. Forget the laser. Could be as "simple" as a massive glowing red or white hot heatsink inefficiently radiating heat away into the vacuum of space (and hopefully not back at Earth) via blackbody radiation. I'll remind you why the sun glows at this point), but you can see the issues.
Actually what I described is basically what a split air conditioning system is but on a space elevator scale. They are structurally arranged the same way for the same reason. The split prevents heat of the machinery from feeding back to the area you are trying to cool.
A: Your proposal is to increase the effective albedo of the Earth, and in principle it could work. No need for the giant lasers though—probably more efficient and much cheaper just to use mirrors.
A: Nice idea... but lasers get their power from electricity.  If we had lots of spare electricity, we could stop generating more and reduce the energy output from coal or gas powered power stations.
So, yes it would be good for the planet to use that electricity to heat our homes etc...but no point really shooting it off into space.
A: From comments:

How would the lasers be powered? – DJohnM


Either with energy gathered from renewable sources or with nuclear power –  DJG

The answer very much depends on the source of energy.
Shooting lasers into the space will transfer energy out from Earth.
For example let's consider a nuclear plant using $^{235}$U. $^{235}$U stores energy in form of bonding energy of the nucleus. By using uranium in a plant, you release this energy. Part might be used for your laser, part will be inevitably dissipated in form of heat on Earth. So the immediate result will be heating up the planet.
In turn, if don't use $^{235}$U in a power plant, it will decay by itself with a half-life of 703.8 My, so eventually it will heat up the planet. Using a uranium-based plant in your application could help to cool the planet on a billion-year time scale, which is rather too long to help with climate changes.
For every other source of energy you would need to redo similar analysis.
