Can a bridge be removed once cemented into the mouth?

Yes, a bridge can certainly be removed. However, are you really asking whether it can be removed or can it be removed and then recemented onto the teeth?If the dentist is going to remove the bridge to replace it with another, then it is usually drilled off. However, if he is trying to take it off to check or change something, and then wanting to recement it, then the technique is to try to tap it off with an instrument. Although that technique is sometimes successful, it also runs the risks of injuring the underlying tooth and even fracturing it off. I do not use that technique in my office because of the high risk of breakage.However, there is another approach that has a lower rate of potential damage. The dentist will have you bite down on a warmed soft wax-like wafer, then cool it down with water. When it is cooled down, it is hard, and sticks to both the bridge and opposite teeth. Then the dentist will ask the patient to open his/her mouth quickly. Sometimes the bridge will come off, but on occasions it can loosen other crowns off of their underlying teeth. Clearly a better way to go than trying to tap off the bridge, but safest approach is for the dentist to just drill off the bridge and make you another one.

Once a bridge is cemented into the mouth, it is highly unlikely it will come out on its own, especially if it's bonded rather than cemented (there are 2 different techniques).  Typically if a bridge has an underlying metal or zirconia framework with porcelain on top it will be cemented, but if it is made of all porcelain or emax (lithium disilicate), then it can be placed using a bonding or adhesive technique.  If it's necessary and it needs to be removed due to decay, poor fit or esthetics, then it is easiest for the doctor to section/cut it with a drill in segments.  A metal framework bridge will typically come off the easiest.  Zirconia and emax will take a little more time. 

However, it would probably be destroyed in the removal process if it was permanently cemented.

I've practice dentistry for 35 years and have encountered this problem many times.  Trying to tap a bridge off often damages the tooth underneath which compromises making a new one or repairing the old.  If a bridge is properly cemented onto the teeth, my recommendation is to not try to remove it, but rather just make a new one.  It would be much safer.  You might even consider placing implants.