What happens if one or more of the magnets in the LHC lose power/fail while it is in full operation?

If anything goes even the slightest bit funny, the entire beam will get dumped--run off into something safe. Then you have to start all over and it takes hours to get running again (assuming the problem wasn't permanent). If the problem was serious, the experiment has to be shut down indefinitely and you have to wait for the hot things to cool down and the cold things to warm up before you can fix it. And then a long time to cool down again. I'm sure this will happen at some point.

http://en.wikipedia.org/wiki/Beam_dump

Vincent, that energy you quoted is the energy carried by each and every proton. The entire beam carries a hell of a lot of energy and if it were uncontrolled and hit something it shouldn't it would do some serious damage. Hence, we have many super-redundant safety systems that will dump the beam at the smallest sign of trouble.

If the magnets are not there to coral the beam, the beam will dissipate and hit the containment chamber wall.

What effect will it have? About the same as a pea shot hitting a battleship. The LHC beam has a power of 7.54 TeV, which is a LOT for a few sub-atomic particles, but works out to 1.2 millionth of a Joule. That could kill a mosquito, maybe.

Edit: Becki above is right about the energy being that of single proton. However, without the confining magnets, the beam would disperse, and spread its energy over a larger area (the total energy of the beam is compared to that of a train running at 160 kph). Evidently, the LHC is meant to survive this but, and that is the point I wanted to make, no harm will come to objects outside of the LHC. The LHC is perfectly safe environmentally.