Well, if you want the opinion of an official source, go to safewateronline.com. According to this website, Qatar ranks 4/5 for its tap water, which means:

"Main cities have generally safe water supplies, which are monitored and regularly tested by an independent board, but level of service of rural and small towns systems could be substandard. Prevalence of waterborne diseases is low."

So that's not as good as other Western countries like USA, UK, or Germany, though better than Saudi Arabia or Oman. I interpret that to mean that water is more than safe for brushing your teeth and cooking. As for drinking, it's probably totally safe as well, at least in Doha. If you still have a mental block against drinking it consider contacting a bottling company which will be the cheapest way to get it.