My one on one student studies public health at NTU. I asked her about the water quality of Taipei. She said one of her professors, one of the foremost authorities on the subject of drinking water, said the water is drinkable from the tap.
The Taipei Water Department also claims this on their website:
As you know, most people boil the tap water to 'clean' it. She said that's a bunch of bullshit. She said while it would kill amoebas in the water that don't even harm your digestive system, it would not boil out the heavier stuff like metals in the water. so essentially whether you boil or not you're drinking virtually the same thing. This puts a big dent in the argument of some locals that the 'pipes are old and dirty' belief.
So I'm thinking that boiling the water is one of these strongly held beliefs/customs of Taiwanese. You can add this one to taking a shower at night instead of in the morning (or both for that matter).
My student mentioned though, for those contact wearers, that it is important to boil out the amoebas if you are using water with your contacts. Amoebas, it seems, are bad for your corneas...