We don't actually know for certain.

We know that early in the war, Hitler sported a Kaiser Mustache (http://en.wikipedia.org/wiki/Wilhelm_II,_German_Emperor just less exaggerated) as seen in photographic evidence, but we know that by 1919 he had the "toothbrush mustache" he is so well known for.

Hitler's defining mustache was fairly common in his day- http://en.wikipedia.org/wiki/File:Chaplin_Kid_Auto_Races.jpg- and was conceived for reasons entirely unrelated to chemical warfare. It was first thought of in the 19th century by Americans looking for a simple, low maintenance 'stache, and not for it's virtues relating to chemical warfare. Hitler likely kept it after the war because it was quite popular in Germany in the period.

Where the debate arises is a point of conflict- Alexander Moritz Frey, who served with Hitler during the Great War, is on the record as saying that Hitler was ordered to trim is mustache because it was posing a hazard to getting his gas mask on. On the other hand cultural historian Ron Rosenbaum claims that Hitler didn't adopt it till 1919. Hitler's sister-in-law also claimed that she was the one responsible for him trimming it as early as 1912 or 1913, despite photographic evidence suggesting otherwise.

.....and that was far more than I ever thought I'd write about mustaches.