It changed, from the best of my understanding, around WWI when the use of gas attacks became a very real threat.

In order for a gas mask to seal properly to your face, which is how it works, it has to make good contact with the skin. Facial hair would obstruct this, leaving one vulnerable. While outlawed by the Geneva convention, there is still the threat of gas attacks, such as against American troops in Iraq.

For similar reasons, soldiers are required to keep their finger nails trimmed, so traces of a gas/contamination doesnt collect and wind up in an eye or nose once the mask is removed.