Is it advisable to shift both the front and back gear simultaneously? Can this cause any damage?
There's nothing wrong with doing this. It works well to shift the rear 1 or 2 shorter when shifting the front 1 taller (and vice versa) to avoid big steps in gearing.
You absolutely can shift front and back simultaneously. However, there is only one scenario where it makes sense and it does require that your mechs to be in tip-top working order and some level of finesse on your part. Moreover, it might only apply to modern Campy derailleurs (don't know about sram, shimano).
The one scenario where it makes sense is if you have rapidly come to the bottom of a steep hill where you KNOW you're going to HAVE TO be in your small chainring. In that situation, I will use both my thumbs to simultaneously thrust the chain to the small chainring and to a small cog (on campy you can sweep across more than one cog with a shift). Doing this will keep you from suffering a huge discontinuity in cadence and you'll be ready to shift one at a time to bigger cogs before your cadence is about to drop. Everyone I know who uses campy does this type of shift at the bottom of very steep hills.
This is good to do because shifting the chainring is somewhat dicey under high torque and low cadence. Better to do it earlier rather than later, and if you are doing it earlier, you'll ALSO want to be in a smaller cog initially so you don't mess up your cadence.
The simple answer is, no, it is not advisable, and yes it can cause damage. The chain uses tension to shift. Shifting with both front and rear derailleurs at the same time will remove that tension. The answer above is absolutely correct, but they are a bit too phlegmatic about the risks involved. Drop a chain due to poor shifting habits, and you risk wrapping the chain in the derailleur. This can and often does pull the derailleur around into the wheel.
In that case, you have lost the derailleur and chain, and you may have lost the wheel. You may even lose the frame if you are unlucky.
There is no real benefit to shifting that way, as it will likely slow down your shifting response, (because of the aforementioned loss of tension on the chain), so why would you do it?
Per the request by @amcnabb, I'm updating this answer to include instruction on the proper way to to shift both your front chain ring and rear chain ring simultaneously.
Simple answer is, you don't. You shift one and then the other. Depending on your need (large gear change, or small gearing change) then you choose the front or rear to shift first. Shift, that one, then if necessary, ass soon as the chain settles on the new gear you've chosen, shift the other.
This is a mechanical necessity. It has nothing to do with choice, preference, or best practice. Your chain requires a certain amount of tension to shift properly. If you shift both the front and rear gears at the same time, then you will not have any tension on the chain shifting down or too much if shifting up. That will cause poor, slow, or no shifting, at best, and at worst, it will cause serious damage to your bike.
With practice and timing, you can make near simultaneous shifts, which will function properly.
The best practice, of course, is to plan ahead, use a gear cluster which takes into account the terrain you intend to ride, and to plan your necessary shifts ahead of time, so that major gear jumps are not ever required. This is safer mechanically, and makes for a more enjoyable ride.
I hope that helps.
Just to build upon the points of view already presented, I think that shifting at the same time would not be a big problem provided that:
I agree with Jay Bazuzi, I also do this "one front, two back", have done it for many years, know a lot of guys who do it, and we have no problem doing this.
BUT I have also seen stories, and being myself a victim, of the problems describes by Zenbike, crushing derailers, chains and hangers by literally pedalling them off after they had jammed.
Only the explanation differs, since I suppose the main problem is shifting under pedal load, and not shifting with both derailers at the same time. Actually, even shifting under load with only one derailer could bring those problems.
Of course, this is only one more theory, and each one should consider the essence of the problem, the facts and theories presented, and check them against its own observations, never forgetting that most probably there are many other factors involved.
Depending on the tension of the chain, you might be slightly more likely to jump, e.g. going from big to little on both front and back at the same time is increasing the slack in the chain and moving it in both directions (front to the left, back to the right).
So depending on the timing, not only will the slack need to be damped out and the indexing might not be as accurate as you'd like, you're also possibly pulling the chain from one end, away from its target at the other.
I don't think is damaging per se, but a jumped chain is to be avoided if you can help it.