Why shouldn't we shine laser light back into the laser? After every laser there is an optical isolator that ensures the laser light cannot re-enter the laser, if reflected and sent back.
What would actually happen if this were to happen? Would the laser break because of too much power?
 A: Its not true that feedback is detrimental to all laser systems.  In fact, some laser systems rely on it heavily for frequency control.  Specifically, diode laser systems in the Littrow Configuration use feedback from a diffraction grating in order to stimulate emission in the laser cavity at the wavelength specific to the first order of diffraction.  Tuning the angle of the grating with a piezo electric device allows for precise frequency control of the laser.  Other diode lasers, such as AR coated diode lasers rely on optical feedback in order to have enough light in the cavity to lase.
With that being said, optical isolators are still a good idea after the laser head output because back reflections from the optics downstream can cause optical feedback that cause the laser to lase in a mode that you don't want.  This can produce laser frequency noise and mode hops which will ruin the operation of the laser, but won't necessarily cause physical damage.  However, some diode chips have a maximum power threshold, and too much feed back (perhaps direct feedback with a mirror and the laser at full current) can damage the chip.
A: If the beam is amplified which is the the beam that emerges a laser amplifier then it is intense enough to damage the optics inside the system. Isolators are usually used for amplifiers then.
But for an oscillator, usually it is not a big issue for short-time period. However since you are sending back the beam it acts like noise for the laser's active material, specially for pulsed lasers.
A: As in the other answers, the reasons for preventing feedback can be
1) to avoid spurious amplification and possible damage, and 
2) to avoid intensity instabilities
The second point is especially important in applications where low intensity noise is required. Dynamically, having feedback means that your system is formed by two coupled cavities, one active, the other passive. In the differential equation for the electric field $\dot{E}(t)=$ (...), feedback can be model as a term $+E(t-\tau)$, where $\tau$ is the roundtrip of the external cavity (the one that gives feedback). This makes the field at $t$ dependent on the field at $t-\tau$, and the dimension of the system goes to infinite (the field at $t-\tau$ depends on the field at $t-2\tau$ and so on). This allows all kind of dynamical instabilities to happen, including hopping, chaos, and excitability. You can find more info looking for Lang Kobayashi (for semiconductor lasers I guess).
While all these behaviors are very interesting for people in non linear dynamics, for engineers and most of the users it is often a phenomenology to avoid.
A: Since the laser beam is focused on the spot of generating unit...it will produce more heat and will cause the crystal to damage..
